![Phase diagram of BaZr x Ti 1-x O 3 bulk ceramic [9].](https://www.researchgate.net/profile/Muhammad-Usman-28/publication/270853984/figure/fig7/AS:614066720632840@1523416323275/Phase-diagram-of-BaZr-x-Ti-1-x-O-3-bulk-ceramic-9_Q320.jpg)
![Direct interactions between stress (σ) and strain (ε), electric field (E) and polarization (P), and magnetic field (H) and magnetization (M), are illustrated with the red, yellow, and blue arrows, respectively. In a single phase multiferroic magnetoelectric material (green arrows), electric field is directly coupled to magnetic field. In many multiferroic magnetoelectric devices, straincoupling (black arrows) between magnetostrictive and piezoelectric phases provides the magnetoelectric effect [109].](https://www.researchgate.net/profile/Muhammad-Usman-28/publication/270853984/figure/fig8/AS:614066720612377@1523416323397/Direct-interactions-between-stress-s-and-strain-e-electric-field-E-and_Q320.jpg)
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| iii
This work is submitted as a dissertation
In partial fulfillment of the requirement for the degree of
DOCTOR OF PHILOSOPHY
IN
PHYSICS
To The
Department of Physics
Quaid-i-Azam University
Islamabad, Pakistan
2015
| iv
CERTIFICATE
This is to certify that the experimental work in this dissertation has been carried
out by Mr. Muhammad Usman under my supervision in the Superconductivity and
Magnetism Lab, Department of Physics, Quaid-i-Azam University, Islamabad,
Pakistan.
Supervisor:
Dr. Arif Mumtaz
Associate Professor
Department of Physics
Quaid-i-Azam University
Islamabad, Pakistan.
Submitted Through:
Prof. Dr. Arshad Majid Mirza
Chairman
Department of Physics
Quaid-i-Azam University
Islamabad, Pakistan.
| v
The thesis titled
“Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y
System” by Muhammad Usman has been reviewed and approved by the following:
Dr. Vladimir V Shvartsman
Senior Researcher
Institut für Materialwissenschaft,
Universität Duisburg-Essen,
Universitätsstrasse 15, D 45141 Essen, Germany.
Email: vladimir.shvartsman@uni-due.de
Prof. Dr. John H. Weaver
Department of Materials Science and Engineering,
University of Illinois at Urbana Champaign,
1304 West Green Street Urbana, Illinois 61801 USA.
Email: jhweaver@illinois.edu
Dr. Antonio Fete
Senior Lecturer
Sheffield Hallam University,
City Centre Campus, Sheaf Building, office #4112,
Street Sheffield, S1 1WB
United Kingdom
Email: a.feteira@shu.ac.uk
Dr. Arshad Saleem Bhatti
Professor
Dean Faculty of Science
Department of Physics
COMSATS Institute of Information Technology
Park Road, Chak Shahzad Islamabad, Pakistan
Email: asbhatti@comsats.edu.pk
Dr. Dr. Muhammad Sabieh Anwar
Associate Professor
Chairman
Department of Physics
School of Science and Engineering
Lahore University of Management Sciences (LUMS)
Lahore, Pakistan.
Email: sabieh@gmail.com, sabieh@lums.edu.pk
| vi
DEDICATION
To
My Family
Especially
) To the sweet memory of my mother.
) To my father whose constant support and
encouragement in my education served as
an inspiration to me to go the highest
possible level.
| vii
ACKONWLEDGEMENTS
“Where there is a will, there is a way.” & “Interdependence is a higher value than independence”
All praise be to Allah, the Cherisher and Sustainer of the worlds, the most beneficent and
merciful, who gave me strength to achieve what I wanted to and blessed me all the ways. Many
peace and blessings be upon the Seal of the Prophet Muhammad, the most perfect and splendid.
I take this opportunity to acknowledge all those who had been instrumental in the success of my
work directly or indirectly.
Foremost, I would like to express my deep sense of gratitude to my advisor Dr. Arif Mumtaz for
his insightful guidance, persistent support, lively discussions, enthusiastic encouragement, expert
advice, constructive and honest criticism and friendly supervision throughout the course of my
doctoral work. I am greatly indebted to him for providing me excellent freedom to work on what I
was interested in. I am really grateful to him for listening to my ideas with such interest and
patience and guiding me with his precious knowledge. He always welcomed new ideas and
encouraged trying different things. I feel so lucky to have worked with him. His enthusiasm and
integral view on research and his mission for providing only high quality work have made a deep
impression on me. He has been excellent mentor and very supportive throughout the work. He is
indeed a remarkable supervisor.
I feel honor to express my gratitude to our group leader Prof. Dr. S. K. Hasanain for his
guidance, generous assistance and support from initial to final level enabled me to develop an
understanding of the subject. Throughout my work, he provided me encouragement, sound
advice, and good teaching. I thank him for his inspiring and encouraging way to guide me to a
deeper understanding of scientific work, and his invaluable comments during the formation of
this dissertation.
I would like to express my deepest gratitude to my research colleague Sobia Raoof for her
remarkable support and encouragement during samples processing, dielectric measurements and
data analysis. To answer her long quires of questions, I learned a lot. She is no doubt a best
companion during this work.
Thanks also to Naveed-ul-Haq, who had to suffer the proof-reading of this work. I am certain
that there is no typing error left.
I would like to thank all former and present members of our research group for all their help,
support and encouragement, and for being great colleagues. Here I want to mention two names:
Tariq, with him I enjoyed the work on polymer blend, vacuum furnace design for the blend
processing and SPM techniques; and with Wafa, the work on magnetic measurements of non-
| viii
interacting magnetic nanoparticles dispersed in polymer was a great fun. It’s a pleasure and
honor for me to work with such a talented people during my doctoral course.
I am also thankful to my Lab fellows Aftab Bhai, Saif Bhai, Shahzad, Naveed, Asad,
Tayyaba and Irfan Ullah for the moral support and for creating a comfortable working
environment.
I am extremely grateful to the staff members of Physics department, Machine shop for their
prompt service and cooperation. I also take this opportunity to acknowledge the prompt services
rendered by the staff of central work shop.
It would be unfair if I don’t extend my admiration to three special people outside the lab and
country, Qaisar (KTH Sweden), Shams (Oxford University London) and Nadeem (ABL
Berkeley), for their support and encourage. Whenever I need research papers not accessed here in
QAU Islamabad, they always welcome me, and send the papers immediately. Thanks to all of you
for help and courage.
I would like to sincerely thank Quaid-i-Azam University Islamabad for providing me the
financial supports under the grant of “Development and Study of Magnetic Nanostructures”
from Higher Education Commission of Pakistan and facilities to carry out my research work.
Thanks to my family, my father, who listened to every tiny detail that I had to tell him about my
work and always encouraged me. The valuable instructions he gave from the mechanical point of
view in troubleshooting during my work of construction are always helpful. He always gives me
great confidence and makes me feel as if I am able to accomplish anything in the world that I
want. Special tribute to my sweet mother (late), her guidance and advices will always serve as
beacon of light throughout the course of my life. I am also thankful to my brother Asim, sisters
(Sobia, Misbah and Sadia) and brother in law Farrukh for care, love, moral support and
motivation.
Last but not least, I thank every person in my life who have always been there for me in every
thick and thin situation and supported me making my PhD possible.
Muhammad Usman
| ix
List of Publications
a) Papers Included in This Thesis
Part of the work presented in this thesis has been published or submitted to refereed
journals. The thesis could have been written as a continuous text based on both published
and unpublished results. To save the reader for repetition of the detailed discussion of the
articles he/she may have read already, the following articles are briefly summarized in the
report and are included as appendices to the thesis:
1. Response to the Comment on “Order Parameter and Scaling Behavior in
BaZrxTi1-xO3 (0.3 < x < 0.6) Relaxor Ferroelectrics”.
Muhammad Usman, Arif Mumtaz, Sobia Raoof and S. K. Hasanain.
Applied Physics Letters 104, 156103 (2014)
My contribution: The initial write up of response to the comment.
2. Order Parameter and Scaling Behavior in BaZrxTi1-xO3 (0.3 < x < 0.6) Relaxor
Ferroelectrics.
Muhammad Usman, Arif Mumtaz, Sobia Raoof and S. K. Hasanain.
Applied Physics Letters 103, 262905 (2013)
My contribution: The calculation of order parameter and discovery of scaling behavior in
BaZrxTi1-xO3, complete dielectric characterization, main manuscript writing and drafting.
3. Magnetic Control of Relaxor Features in BaZr0.5Ti0.5O3 and CoFe2O4 Composite.
Muhammad Usman, Arif Mumtaz, Sobia Raoof and S. K. Hasanain.
Applied Physics Letters 102, 112911 (2013)
My contribution: Sample preparations, complete dielectric characterization, discovery of
magnetic field control of the relaxor behavior in composites, data analysis and calculation
of order parameter, main manuscript writing and drafting.
b) Additional Output PhD Course
In addition to the included papers, I contributed to the papers listed below that have been
published during my PhD project. These are not included in this thesis.
4. Static Magnetic Properties of Maghemite Nanoparticles
Zulfiqar, Muneeb Ur Rahman, M. Usman, Syed Khurshid Hasanain, Zia-ur-
Rahman, Amir Ullah and Ill Won Kim
Journal of the Korean Physical Society, 65, 1925 (2014)
My Contribution: My contribution in this paper is the measurements Magnetic
measurements using Physical Properties Measurement System (PPMS) and analysis.
5. Surface Defects: Possible Source of Room Temperature Ferromagnetism in Co-
Doped ZnO Nanorods
N. Tahir, A. Karim, K. A. Persson, S. T. Hussain, A.G. Cruz, M. Usman, M. Naeem,
R. Qiao, W. L. Yang, Y.-D Chuang, Z. Hussain.
Journal of Physical Chemistry C 117, 8968 (2013)
| x
My Contribution: This collaborative project with Nadeem Tahir and his co-workers was
partially carried out in our lab. My contribution in this paper is the measurements of the
diffuse reflectance spectra using Perkin Elmer Lambda 950 Photo-spectrometer and
Magnetic measurements using Physical Properties Measurement System (PPMS) and
analysis of the data.
6. Synthesis, Structural, and Magnetic Characterization of Mn1−xNixFe2O4 Spinel
Nanoferrites.
Ahmed Faraz, Mudasara Saqib, Nasir M. Ahmad, Fazal-ur-Rehman, Asghari
Maqsood, Muhammad Usman, Arif Mumtaz, Muhammed A. Hassan
Journal of Superconductivity and Novel Magnetism 25, 91 (2012)
My Contribution: This collaborative project with Prof. Nasir M. Ahmad and his student
Ahmed Faraz was carried out in our lab. The magnetic measurements were performed by
me using vibrating sample magnetometer.
7. Magnetic study of Cu1−xMnxO (0 ≤ x ≤ 0.08) nanoparticles.
Shahzad Hussain, A. Mumtaz, S. K. Hasanain, and M. Usman
Journal of Applied Physics 111, 023908 (2012)
My Contribution: My contribution in this paper is the measurements Magnetic
measurements using Physical Properties Measurement System (PPMS).
8. Effect of Crystallographic Texture on Magnetic Characteristics of Cobalt
Nanowires
K. Maaz, S. Karim, M. Usman, A. Mumtaz J. Liu, J. L. Duan, M. Maqbool
Nanoscale Research Letters 5, 1111 (2010)
My Contribution: This collaborative project with Dr. Maaz Khan was carried out in our
lab. My contribution in this paper is the measurements of X-ray diffraction and Magnetic
measurements using Physical Properties Measurement System (PPMS).
9. Magnetic response of core-shell cobalt ferrite nanoparticles at low temperature.
K. Maaz, M. Usman, S. Karim, A. Mumtaz, S. K. Hasanain M. F. Bertino
Journal of Applied Physics 105, 113917 (2009)
My Contribution: Data analysis of magnetic measurements partially done by me.
10. Effect of vanadium doping on structural, magnetic and optical properties of ZnO
nanoparticles
N. Tahir, S.T. Hussain, M. Usman, S.K. Hasanain, A. Mumtaz
Applied Surface Science 255, 8506 (2009)
My Contribution: This collaborative project with Nadeem Tahir and his co-workers was
carried out in our lab. XRD, Magnetic measurements using PPMS, UV/Vis optical
measurements and data analyses.
| xi
ABSTRACT
In this thesis we report the ferroelectric relaxor behavior of the zirconium doped
barium titanate BaZrxTi1-xO3 solid solutions and their composite with Cobalt ferrite.
BaZrxTi1-xO3 ceramics of the several compositions (0 ≤ x ≤ 0.8) have been prepared by
conventional solid state synthesis route. Dielectric behavior of the BaZrxTi1-xO3 ceramics
has been studied in the temperature range from 300 K to 10 K at various frequencies in
the range 0.2-500 kHz. It was observed that for composition where x ≥ 0.3 the dielectric
response show a diffused dielectric transition with decreasing temperatures. Furthermore
the dielectric peak shows a strong frequency dependence moving towards higher
temperatures for increasing frequencies. We discuss dependence of the dielectric constant
on temperature, frequency and concentration in term of correlations among the polar nano
regions. By considering the relaxor ferroelectrics as an analogue to classical magnetic
spin glass system, we have analyzed the relaxor behavior within the mean field theory by
estimating the Edward-Anderson order parameter EA
q. We find that EA
q calculated for the
different concentrations obeys a scaling behavior
()
n
mEA TTq /1 −= , where Tm are the
respective dielectric maxima temperatures and n = 2.0 ± 0.1. The frequency dependence
of the EA
q is also consistent with the above mentioned picture.
Magnetoelectric composites were prepared using conventional sintering.
BaZrxTi1-xO3 was used as the relaxor ferroelectric material and the ferrite CoFe2O4 was
used as the magnetostrictive component. We report the effect of magnetic field on the
dielectric response in the relaxor ferroelectric and ferromagnetic composite
[BaZr0.5Ti0.5O3]0.65:[CoFe2O4]0.35. Relaxor characteristics such as dielectric peak
temperature and activation energy show a dependence on applied magnetic fields, viz. the
dielectric peak shifts to higher temperature with increasing magnetic field. This is
explained in terms of increasing magnetic field induced frustration of the polar nano
regions comprising the relaxor. The results are also consistent with the mean field
formalism of dipolar glasses. It is seen that the variation of the spin glass order parameter
EA
q(T) with applied magnetic fields is consistent with an increased frustration and earlier
blocking of polar nano regions.
| xii
Contents
CERTIFICATE ................................................................................................................................................ iv
DEDICATION.................................................................................................................................................. vi
ACKONWLEDGEMENTS ........................................................................................................................... vii
List of Publications .......................................................................................................................................... ix
a) Papers Included in This Thesis ........................................................................................................... ix
b) Additional Output PhD Course ........................................................................................................... ix
ABSTRACT ...................................................................................................................................................... xi
Contents ........................................................................................................................................................... xii
List of Figures .................................................................................................................................................. xv
List of Tables ................................................................................................................................................ xviii
Thesis Organization ....................................................................................................................................... xix
CHAPTER 1 INTRODUCTION & BACKGROUND ................................................... 1
1.1 Introduction ............................................................................................................................................ 1
1.2 Dielectrics ................................................................................................................................................ 2
1.2.1 Dielectric Polarization ..................................................................................................................... 2
a) Mechanism of Polarization ............................................................................................................. 2
1.2.2 Dielectric Constant and Frequency Dispersion .............................................................................. 4
1.2.3 Crystalline Material Classifications ................................................................................................ 5
1.3 Crystal Structures .................................................................................................................................. 6
1.3.1 Perovskite Structure ........................................................................................................................ 7
1.3.2 Spinel Ferrites .................................................................................................................................. 8
1.3.3 Spinel-Perovskite Composites ........................................................................................................ 9
1.4 Ferroelectrics .......................................................................................................................................... 9
1.4.1 Ferroelectric Phase Transition ...................................................................................................... 10
a) First and Second Order Phase Transitions .................................................................................... 10
1.4.2 Ferroelectric Hysteresis ................................................................................................................. 12
1.4.3 Displacive and Order Disorder Type Phase Transitions .............................................................. 12
1.4.4 Barium Titanate ............................................................................................................................. 13
1.5 Antiferroelectrics ................................................................................................................................. 15
1.6 Diffuse Phase Transition ..................................................................................................................... 16
1.7 Relaxor Ferroelectrics ......................................................................................................................... 17
1.7.1 Macroscopic Properties of Relaxor Ferroelectrics ....................................................................... 18
1.8 Theoretical Models .............................................................................................................................. 20
1.8.1 Compositional Fluctuation Model ................................................................................................ 20
1.8.2 Superparaelectric Model ............................................................................................................... 21
1.8.3 Dipolar Glass Model ..................................................................................................................... 21
1.8.4 Vogel-Fulcher Law ....................................................................................................................... 22
1.8.5 Random Field Model ..................................................................................................................... 22
1.8.6 Spherical Random Bond Random Field Model ............................................................................ 23
1.8.7 Edward Anderson Order Parameter .............................................................................................. 23
| xiii
a) Reconstruction of Edward-Anderson Order Parameter for Relaxors .......................................... 24
1.9 Temperature Evolution of Polar Nano Regions ............................................................................... 27
1.10 BaTiO3 Based Relaxor System ........................................................................................................... 29
1.10.1 BaZrxTi1-xO3 Solid Solution .......................................................................................................... 31
1.10.2 Phase Diagram of BaZrxTi1-xO3 ..................................................................................................... 31
1.11 Multiferroics ......................................................................................................................................... 32
CHAPTER 2 MOTIVATIONS & AIMS ....................................................................... 35
CHAPTER 3 EXPERIMENTAL METHODOLOGY ................................................. 38
3.1 Experimental Procedures .................................................................................................................... 38
3.2 Fabrication of Ferroelectric and Multiferroics Materials .............................................................. 39
3.2.1 Furnaces ......................................................................................................................................... 40
a) Box Furnace ................................................................................................................................... 40
b) Tube Furnace ................................................................................................................................. 40
3.3 X-ray Diffractometer ........................................................................................................................... 40
3.3.1 Powder Samples Configuration .................................................................................................... 41
3.3.2 Thin Film Configuration ............................................................................................................... 42
3.4 Dielectric Properties Measurement ................................................................................................... 43
3.4.1 Cryostat .......................................................................................................................................... 43
3.4.2 Sample Holder ............................................................................................................................... 44
3.4.3 Temperature Controller (331 LakeShore) .................................................................................... 45
3.4.4 Dielectric Constant ........................................................................................................................ 45
3.4.5 Dielectric Loss ............................................................................................................................... 46
3.4.6 High Temperature Dielectric Measurement System .................................................................... 47
a) Description of the Set-up .............................................................................................................. 47
a) Heating Element ............................................................................................................................ 49
b) Sample holder ................................................................................................................................ 49
c) Vacuum Tight Connector .............................................................................................................. 50
d) Temperature Controller (ITC502 Oxford) .................................................................................... 50
3.5 Vibrating Sample Magnetometer ....................................................................................................... 51
CHAPTER 4 PROCESSING AND STRUCTURAL CHARACTERIZATION ........ 52
4.1 Synthesis Techniques ........................................................................................................................... 52
4.2 Solid State Reaction Method .............................................................................................................. 53
4.2.1 Precursors ...................................................................................................................................... 53
4.2.2 Weighing, Mixing and Grinding ................................................................................................... 54
4.2.3 Calcination ..................................................................................................................................... 54
4.2.4 Green Body Formation .................................................................................................................. 55
4.2.5 Sintering......................................................................................................................................... 55
4.2.6 Electroding .................................................................................................................................... 55
4.3 Synthesis of Relaxor Ferroelectrics Ba(Zr,Ti)O3 ............................................................................. 56
| xiv
4.4 Synthesis of Magnetic Ferrites CoFe2O4 ........................................................................................... 58
4.5 Synthesis of Composite Ceramics [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y ..................................................... 59
4.6 Structural Analysis .............................................................................................................................. 61
4.6.1 Structural Characterization of Ba(Zr,Ti)O3 .................................................................................. 61
4.6.2 Structural Characterization of CoFe2O4 ........................................................................................ 66
4.6.3 Structural Characterization of [Ba(Zr,Ti)O3]0.65:[CoFe2O4]0.35 .................................................... 67
CHAPTER 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM
TITANATE ................................................................................................ 70
5.1 Dielectric Spectroscopy ....................................................................................................................... 71
5.1.1 Temperature Dependent Dielectric Spectra of BaTiO3 ................................................................ 72
5.1.2 Temperature Dependence of Dielectric Spectra of BaZr0.3Ti0.7O3 ............................................... 74
5.1.3 Reduced Dielectric Spectra of BaZrxTi1-xO3 ................................................................................. 75
5.1.4 Dielectric Spectra of BaZrxTi1-xO3 and Curie Weiss Behavior .................................................... 80
5.1.5 Frequency Dispersion and Freezing Behavior in BaZrxTi1-xO3 .................................................... 84
5.1.6 Relaxation Time and Vogel-Fulcher Law .................................................................................... 86
5.2 Edward-Anderson Order Parameter and Scaling Behavior in BaZrxTi1-xO3 .............................. 90
5.3 Summary ............................................................................................................................................... 98
CHAPTER 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS ............... 99
6.1 Magnetic Measurements ................................................................................................................... 100
6.2 Dielectric Spectroscopy ..................................................................................................................... 101
6.1.1 Frequency Dispersion in Dielectric Susceptibility ..................................................................... 102
6.1.2 Dielectric Spectra of BaZrxTi1-xO3 and Curie Weiss Behavior .................................................. 104
6.1.3 Relaxation Time and Vogel-Fulcher Law .................................................................................. 106
6.3 Magnetic Field Effect on the Dielectric Susceptibility of [BaZr0.5Ti0.5O3]0.65:[CoFe2O4]0.35 ...... 108
6.4 Magnetic Field Effect on the Order Parameter of [BaZr0.5Ti0.5O3]0.65:[CoFe2O4]0.35 ................. 110
CHAPTER 7 SUMMARY AND CONCLUSIONS ..................................................... 113
References ...................................................................................................................................................... 117
APENDEX I. PUBLISHED PAPERS INCLUDED IN THIS THESIS ................ 124
| xv
List of Figures
Figure 1.1: Polarization mechanisms in the dielectric materials. .................................................................... 3
Figure 1.2: Real and imaginary part of the dielectric constant as a function of frequency. ............................ 4
Figure 1.3: A classification scheme for 32 point groups of crystallography. ................................................... 6
Figure 1.4: (a) The typical perovskite structure of ABO3. (b) The A cations occupy the larger spaces in the
12-fold oxygen coordinated holes; the B cations occupy the much smaller octahedral holes (six-fold
coordination). Full or partial substitution of the A or B cations with cations of different valence is possible.
When the overall valence of the A-site and B-site cations (n+m) adds up to less than 6, the missing charge
is made up by introducing vacancies at the oxygen lattice sites. ...................................................................... 7
Figure 1.5: (a) Spinel unit cell structure, (b) octahedral interstice (B site: 32 per unit cell, 16 occupied),
and (c) tetrahedral interstice (A site: 64 per unit cell, 8 occupied). In (a) the ionic positions are the same in
octants sharing only one edge and different in octants sharing a face. Each octant contains 4 Oxygen ions.
In (a) ionic positions in only two adjacent octants are shown, where the octant on the left contains
octahedral and the one on the right contains tetrahedral sites. All ions are positioned on body diagonals of
the octants and the octant on the right contains a tetrahedral site at the octant centre. .................................. 9
Figure 1.6: Free Energy F(P), polarization P(T), and permittivity ε(T), ε-1(T) for a first and second order
phase transition for E = 0 [25]. ....................................................................................................................... 11
Figure 1.7: Typical ferroelectric hysteresis loop. ............................................................................................ 12
Figure 1.8: The unit cells for the four phases of BaTiO3. Arrows indicate direction of the polarization. In
the cubic phase a = b = c = 4.009 Å. ............................................................................................................... 14
Figure 1.9: Properties of barium titanate as a function of the temperature: a) unit-cell distortions lattice
dimensions after [29] b) spontaneous polarization c) relative permittivities measured in the a and the c
direction after [34]. .......................................................................................................................................... 14
Figure 1.10: Comparison of ferroelectrics and relaxor ferroelectrics in terms of field-induced polarization
(left), temperature-dependent polarization (middle), and relative permittivity (right) (after Ref. [45]) ....... 19
Figure 1.11: Evolution of PNRs upon temperature decrease [74]. ................................................................ 27
Figure 1.12: The cubic perovskite lattice showing the location of some substituents and vacancies [45]. .. 29
Figure 1.13: Effect of several isovalent substitutions on the transition temperature of BaTiO3 [26]. .......... 30
Figure 1.14: Phase diagram of BaZrxTi1-xO3 bulk ceramic [9]....................................................................... 31
Figure 1.15: Direct interactions between stress (σ) and strain (ε), electric field (E) and polarization (P),
and magnetic field (H) and magnetization (M), are illustrated with the red, yellow, and blue arrows,
respectively. In a single phase multiferroic magnetoelectric material (green arrows), electric field is
directly coupled to magnetic field. In many multiferroic magnetoelectric devices, strain-coupling (black
arrows) between magnetostrictive and piezoelectric phases provides the magnetoelectric effect [109]. ..... 33
Figure 3.1: The overall experimental procedure for the bulk ceramic. .......................................................... 38
Figure 3.2: Various steps in conventional sintering method for processing of bulk materials. ..................... 39
Figure 3.3: Powder sample configuration for XRD measurements. ............................................................... 41
Figure 3.4: Configuration for the phase analysis of thin film. ........................................................................ 42
Figure 3.5: Schematic block diagram of the dielectric measurement setup. .................................................. 43
Figure 3.6: Complete view of setup used for dielectric constant measurements. ........................................... 44
Figure 3.7: System wiring inside closed cycle system model CCS-350. ........................................................ 45
Figure 3.8: Photograph of high temperature dielectric measurement setup. ................................................. 47
Figure 3.9 : The cross sectional view of the vacuum ports for vacuum pump and sample holder probe head.
All the above assembly is made ........................................................................................................................ 48
Figure 3.10: Schematic diagram of the heating element. ................................................................................ 49
Figure 3.11: Sample Holder designed for high temperature dielectric measurements. ................................. 50
Figure 3.12: The commercial vacuum tight pin connector ............................................................................. 50
Figure 4.1: Heat treatment cycle and procedure for magnetic ferroelectric ceramic Ba(Zr,Ti)O3. .............. 57
Figure 4.2 Heat treatment cycle and procedure for magnetic ceramic CoFe2O4 sample. ............................. 59
Figure 4.3: Heat treatment cycle and procedure for composite [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y samples. ....... 60
Figure 4.4: The room temperature X-ray diffractogram of BaTiO3. ............................................................... 62
| xvi
Figure 4.5: The X-ray diffractogram of Ba(Zr,Ti)O3 (0.3≤ x ≤0.8). ............................................................... 63
Figure 4.6: Variation of lattice constant “a” and lattice volume “V” with Zr concentration in the
Ba(Zr,Ti)O3 (0.3≤ x ≤0.8). ................................................................................................................................ 64
Figure 4.7: Variation of XRD density ρXRD with Zr concentration in the Ba(Zr,Ti)O3 (0.3≤ x ≤0.8). ........... 66
Figure 4.8: X-ray diffractogram of CoFe2O4. .................................................................................................. 66
Figure 4.9: X-ray diffractogram of [BZTx]0.65:[CFO]0.35 (0.4≤ x ≤0.7). ........................................................ 67
Figure 4.10: Peak shift in the XRD pattern with the Zr content in the [BZTx]0.65:[CFO]0.35. ....................... 68
Figure 5.1: Dielectric spectrum of BaTiO3 as a function of temperature measured at 500 kHz frequency. . 72
Figure 5.2: The inverse of the dielectric permittivity (1/εʹ) as function of temperature of BaTiO3 sample for f
= 500 kHz. The line shows the fit to the Curie-Weiss behavior. ..................................................................... 73
Figure 5.3: Dielectric spectrum of BaZr0.3Ti0.7O3 as a function of temperature. ........................................... 74
Figure 5.4: Reduced dielectric spectra, a) real part, b) imaginary part of dielectric permittivity for
BaZrxTi1-xO3 with (0.3 ≤ x ≤ 0.8). ..................................................................................................................... 75
Figure 5.5: Randomly distributed polar nano regions in a soft-mode host lattice. (a) At high temperature
(Tm < T < TB) the polar nano regions and the correlation radius, rc are small. (b) Both grow and coalesce
at low temperatures. ......................................................................................................................................... 77
Figure 5.6: Variation in peak temperature of real and imaginary part of ε′ of with Zr concentration. ........ 78
Figure 5.7: Variation in absolute value of ε′m and εʺm with Zr concentration measured at 500 kHz. ........... 79
Figure 5.8: The inverse of the dielectric permittivity (1/εʹ) as function of temperature for various
concentrations, for f = 500 kHz. The lines are the fit to the Curie-Weiss behavior. The deviation
temperature Tdev and the x-intercept θ are marked for the BZT0.6. ................................................................ 81
Figure 5.9: Plot of ln(1/εʹ-1/εʹm) vs. ln(T-Tʹm) of the data taken at 500 kHz applied frequency. The lines are
the fit to the Modified Curie-Weiss law (Eq. 5.5). The values of the slope γ are mention for all the BZTx
sample. .............................................................................................................................................................. 83
Figure 5.10: Frequency dependence of dielectric spectra of BaTiO3 ceramic samples. ................................ 84
Figure 5.11: Frequency dependence of dielectric spectra of BZTx ceramic samples. ................................... 85
Figure 5.12: Peak temperature dependence of frequency for BZTx (0.2 ≤ x ≤ 0.8) ceramics. Solid lines
represent the fitting by using the Vogel-Fulcher equation 5.8. ....................................................................... 88
Figure 5.13: The EA order parameter EA
q as a function of temperature for various BZTx compositions and
f = 500 kHz determined using Eq. 5.11. Note that the onset temperature Tdev of EA
q is the same as the
deviation temperature determined from the Curie-Weiss behavior ................................................................ 92
Figure 5.14: EA-order parameter EA
q as a function of scaled temperature T/Tm for various BZTx
compositions and f = 500 kHz. The degree of overlap of the curves below T = 0.85 Tm and the deviations at
higher T are evident. The solid curve indicates the scaling function
()
n
mEA TTq /1 −= with n = 2.05 ± 0.1. 93
Figure 5.15: EA-order parameter EA
q for various compositions BZTx as function of reduced temperature
T/Tm measured at frequencies of 0.2-500 kHz.................................................................................................. 95
Figure 5.16: Frequency dependence of the onset temperature T*/Tm of EA-order parameter EA
q for various
compositions BZTx. ........................................................................................................................................... 97
Figure 5.17: Variation in change in onset point and change in EA
q measured at 0.2 kHz and 500 kHz
frequency as function of Zr content. ................................................................................................................. 97
Figure 6.1: Magnetic hysteresis curves of samples measured at room temperature. ................................... 100
Figure 6.2: Dielectric spectrum of (BaZr0.5Ti0.5O3)0.65 : (CoFe2O4)0.35 measured at 500 kHz. .................... 102
Figure 6.3: Temperature dependent dielectric spectra (εʹ, εʺ ) of the (a) BZT0.5, (b) BZ65C35 (H = 0 kOe),
(c) BZ65C35 (H = 5 kOe) and (d) BZ65C35 (H = 7 kOe). ........................................................................... 103
Figure 6.4: (a-c) 1/εʹ vs. temperature measured at 500 kHz. Straight line shows the Curie-Weiss behavior at
higher temperature. The lines are the fit to the Curie-Weiss law (equation 6.1). (d-f) Plot of ln(1/εʹ-1/εʹm) vs.
ln(T-Tʹm) of the data taken at 500 kHz applied frequency. The lines are the fit to the Modified Curie-Weiss
law (equation 6.3). .......................................................................................................................................... 105
Figure 6.5: ln(f) vs. Tm curves of BZT0.5 and BZ50C35 samples. Lines represent the fitted curve using
Vogel Fulcher law (Eq. 1.11). ........................................................................................................................ 107
| xvii
Figure 6.6: Dielectric spectrum (εʹ, εʺ) of BZ65C35 taken in the presence of
0 kOe,
z
5 kOe and
c
7
kOe magnetic field. Arrow indicates the direction of increasing field. ......................................................... 108
Figure 6.7: Variation in Edward-Anderson order parameter EA
q at different applied magnetic field. Arrow
indicates the direction of increasing applied magnetic field. ........................................................................ 111
| xviii
List of Tables
Table 1.1: Comparison between normal and relaxor ferroelectrics [46]. ...................................................... 19
Table 3.1: Powder sample configuration X-ray beam optics. ......................................................................... 41
Table 4.1: The precursors used for the bulk powder ceramic processing. ..................................................... 54
Table 4.2: The ratio of molecular weight of the precursors for Ba(Zr,Ti)O3. ................................................ 57
Table 4.3: Values of the lattice parameter of Ba(Zr,Ti)O3 calculated from the XRD diffractogram. ............ 64
Table 4.4: Values of the lattice parameter of BaTiO3 and BaZrO3 calculated from data. ............................. 65
Table 4.5: Values of the lattice parameter of [BZTx]0.65:[CFO]0.35 calculated from the XRD diffractogram.
........................................................................................................................................................................... 69
Table 5.1: The Critical parameter (Curie-Weiss constant C, Curie-Weiss temperature θ, the deviation
temperature Tdev, peak temperature Tm and the degree of deviation ΔTm) calculated from the Curie-Weiss
law fit of the experimental data in the high temperature paraelectric regime................................................ 81
Table 5.2: The critical parameters (Modified Curie-Weiss constant C1 and the diffuseness exponent γ)
calculated from the Modified Curie-Weiss law fit of the experimental data in the temperature range above
Tʹm. ..................................................................................................................................................................... 83
Table 5.3: The critical parameters, pre-exponential factor fₒ, the relaxation time τₒ, activation energy Ea
and static freezing temperature TVF calculated from the Vogel-Fulcher law (Eq. 1.11) fit to the experimental
data (peak temperature of real part of dielectric constant Tʹm). .................................................................... 89
Table 5.4: The critical parameters, pre-exponential factor fₒ, the relaxation time τₒ, activation energy Ea
and static freezing temperature TʺVF calculated from the Vogel-Fulcher law (Eq. 1.11) fit to the
experimental data (peak temperature of imaginary part of dielectric constant Tʺm). ................................... 89
Table 6.1: Values of saturation magnetization MS measured at 7 kOe maximum filed, remanent
magnetization Mr and Coercivity HC of the CFO and BZ65C35 samples at room temperature. ................. 101
Table 6.2: The Critical parameter (Curie-Weiss constant C, Curie-Weiss temperature θ, the deviation
temperature Tdev, peak temperature Tm and the degree of deviation ΔTm) calculated from the Curie-Weiss
law fit of the experimental data in the high temperature paraelectric regime.............................................. 105
Table 6.3: The critical parameters (Modified Curie-Weiss constant C1 and the diffuseness exponent γ)
calculated from the Modified Curie-Weiss law fit of the experimental data in the temperature range above
Tʹm. ................................................................................................................................................................... 106
Table 6.4: The critical parameters, pre-exponential factor fₒ, the relaxation time τₒ, activation energy Ea
and static freezing temperature TVF for BZ65C35 obtained from Vogel-Fulcher law (Eq. 1.11) fit for the
peak temperature (Tʹm) variation with frequency. Data taken in presence of applied magnetic fields (0, 5
and 7 kOe). ...................................................................................................................................................... 107
Table 6.5: The critical parameters, pre-exponential factor fₒ, the relaxation time τₒ, activation energy Ea
and static freezing temperature TVF for BZ65C35 obtained from Vogel-Fulcher law (Eq. 1.11) fit for the
peak temperature (Tʺm) variation with frequency. Data taken in presence of applied magnetic fields (0, 5
and 7 kOe). ...................................................................................................................................................... 108
| xix
Thesis Organization
Chapter 1 summarizes the most important and fundamental concepts required for the
description and understanding of relaxor ferroelectrics as discussed in the present work.
The part revisits dielectric, ranging from basis on polarization mechanisms to theories on
relaxor ferroelectrics. Later sections will elucidate materials of importance for this work,
namely barium zirconate titanate. A concise selection of the most essential research
articles are referenced in this chapter.
Chapter 2 describes the motivation and aim of the present work.
Chapter 3 describes the experimental techniques used to characterize bulk ceramic
samples. Structural, electrical and magnetic properties of the polycrystalline bulk can be
determined using various methods. A detailed description about synthesis and
characterization tools used during the course of present work is also given in the chapter.
Chapter 4 gives the detail description about synthesis and structural characterization of
the prepared bulk ceramics.
Chapter 5 deals with the dielectric properties of the BaZrxTi1-xO3 solid solution. I report
the relaxor behavior of the zirconium doped barium titanate BaZrxTi1-xO3 solid solutions
and discuss the temperature, frequency and concentration dependence in terms of
correlations among the polar nano regions. Different models being employed to find the
critical parameter of relaxor system. The relaxor behavior is analyzed within the mean
field theory by estimating the Edward-Anderson order parameter EA
q. I find that the
calculated EA
q for the different concentrations obeys a scaling behavior,
()
n
mEA TTq /1 −=
where Tm are the respective dielectric maxima temperatures and n = 2.0 ± 0.1. The
frequency dependence of the also shows results consistent with the above mentioned
picture.
Chapter 6 deals with the dielectric properties of the [BaZr0.5Ti0.5O3]0.65:[CoFe2O4]0.35
composite material. I report the effect of magnetic field on the dielectric response in a
relaxor ferroelectric and ferromagnetic composite [BaZr0.5Ti0.5O3]0.65:[CoFe2O4]0.35.
Relaxor characteristics such as dielectric peak temperature and activation energy show a
dependence on applied magnetic fields. This is explained in terms of increasing magnetic
field induced frustration of the polar nano regions comprising the relaxor. The results are
also consistent with the mean field formalism of dipolar glasses. It is found that the
variation of the spin glass order parameter EA
q(T) is consistent with increased
frustration and earlier blocking of polar nano regions with increasing magnetic field.
Chapter 7 summarizes and concludes the findings of this work along with the future
direction.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 1
Chapter 1 INTRODUCTION & BACKGROUND
This chapter summarizes the most important and fundamental concepts required for
the description and understanding of the relaxor ferroelectric as discussed in the
present work. This part revisits dielectrics, ranging from basics of polarization
mechanisms to theories of relaxor ferroelectrics. Later sections will elucidate materials
of importance for this work, namely barium zirconate titanate. A concise selection of
the most essential research articles are referenced in this chapter.
1.1 Introduction
Oxides comprise a wide class of materials exhibiting rich crystal structures and
physical properties which make them ideal candidates for different applications. There
has been a recent resurgence of interest in oxides in general and complex oxides in
particular owing to their piezoelectric, ferroelectric, ferromagnetic, ferrimagnetic, and
multiferroic properties, as well as in many cases owing to large dielectric constants.
While these properties had been recognized many decades ago, the renewed interest
stems from exploiting improved processing/characterization techniques, which would
potentially allow not only higher quality materials but also thin films and heterostructures
with new functionalities as well as composite structures with e.g. magnetoelectric
coupling, a concept which has gotten a good deal of attention recently.
Relaxor ferroelectrics belong to a subgroup of ferroelectric materials. Relaxor
ferroelectrics [1] are of great technological importance [2, 3] as they offer a wide range of
useful properties. They exhibit outstanding dielectric, electro-elastic, and electro-optic
properties related to their complex nanoscale structure [4, 5]. The useful properties
include high permittivity (used in capacitors), high piezoelectric effects (used in sensors,
actuators and resonant wave devices such as radio-frequency filters) [6], high pyroelectric
coefficients (used in infra-red detectors), strong electro-optic effects (used in optical
switches), anomalous temperature coefficients of the resistivity (used in electric-motor
overload protection circuits) and slim ferroelectric hysteresis (used in non-volatile
ferroelectric random access memories (NV-FeRAM) devices). Current NV-FeRAM are
made of lead zirconate titanate (PZT), strontium bismuth tantalate, and bismuth titanate
[7]. There are few limitations of these materials, better compounds or mixtures must be
used in future devices, which require a better understanding of the relaxor behaviour.
Hence, it is vital to understand the atomic mechanisms of relaxor ferroelectrics.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 2
This thesis deals with environmental friendly lead free relaxor ferroelectrics and
magnetoelectric composites. In the past, extensive research on ferroelectric relaxor
behavior has been carried out from basic experimental measurements, first principle
theoretical calculations to device applications (e.g. piezoelectric transducers, actuators,
capacitors etc.) ever since the discovery of exceptional dielectric properties of the lead
based disordered perovskites like PbMg1/3Nb2/3O3 (PMN) and PbZn1/3Nb2/3O3 (PZN).
Although numerous relaxor materials have been extensively studied over a long period
the relaxor mechanism always opens up new questions about the unusual ferroelectric
properties of this family of materials. Recently, research emphasis is leaning towards
barium and bismuth based relaxors, as these are environmental friendly lead free oxides,
e.g. Ba(Sn,Ti)O3, Ba(Zr,Ti)O3 [8, 9], bismuth sodium titanate (Bi1/2Na1/2TiO3) and
BaTiO3-BaScO3 [10-12]. On the other hand the coexistence of magnetic and ferroelectric
behavior is hard to achieve. In this thesis novel magnetoelectric composites are designed
and investigated. This chapter intends to provide a brief background of information
relevant to the current investigation.
1.2 Dielectrics
Dielectric materials have a strong influence on the evolution of electrical
engineering, electronics, and information technology. They are typically crystalline or
amorphous inorganic as well as organic compounds and polymers [13]. Dielectrics are
insulating materials that do not conduct electric current due to the very low density of free
charge carriers. Here the electrons are bound to microscopic regions within the materials.
For a pure dielectric material, the polarization (dipole moment per unit volume) is
proportional to the electric field in linear approximation by:
EP eo
r
r
χε
=
where εr is the relative dielectric constant
and χe : dielectric susceptibility 1−= re
ε
χ
1.2.1 Dielectric Polarization
a) Mechanism of Polarization
In general there are four different mechanisms which can contribute to the
polarization of a dielectric material (Figure 1.1). The total polarization of the dielectric
material is the sum of all the following contributions.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 3
i. Electronic Polarization Pelec
Electronic polarization exists in all dielectrics. It is based on the displacement of the
negatively charged electron shell against the positively charged core. The electronic
polarizability αel is approximately proportional to the volume of the electron shell. In
general αel is temperature-independent, and large atoms have a large electronic
polarizability.
Figure 1.1: Polarization mechanisms in the dielectric materials.
ii. Ionic Polarization (Atoms) Pions
Ionic polarization is observed in ionic crystals and describes the displacement of the
positive and negative sublattices under an applied electric field (e.g. NaCl). Ionic
polarization may induce ferroelectric transition as well as dipolar polarization.
iii. Orientation or Dipolar Polarization Por
Orientation polarization describes the alignment of permanent dipoles as a function
of the applied electric field. At ambient temperatures, usually all dipole moments have
statistical distribution of their directions. An electric field generates a preferred direction
for the dipoles, while the thermal movement of the atoms perturbs the alignment. The
average degree of orientation is a function of the applied field and the temperature. The
dipolar or orientational polarizability is given by [14]
Tk
p
B
or 3
2
=
α
1.1
where p is the dipole moment, kB denotes the Boltzmann constant and T is the absolute
temperature.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 4
iv. Space Charge Polarization Psc
Space charge polarization could exist in dielectric materials which show spatial
inhomogeneities of charge carrier densities. The space charge can occur at the
metal/dielectric interface, at grain boundaries in ceramics and domain walls in
ferroelectrics. Thus, interfaces play a crucial role in the space charge effect but this
extrinsic polarization is much more pronounced if charged defects are localized at these
interfaces. Space charge polarization effects are not only of importance in semiconductor
field-effect devices, they also occur in ceramics with electrically conducting grains and
insulating grain boundaries (so-called Maxwell-Wagner polarization).
1.2.2 Dielectric Constant and Frequency Dispersion
The total dielectric constant of dielectric material results from all the contributions
discussed above. The contributions from the lattice are called intrinsic contributions, in
contrast to extrinsic contributions.
4342143421
Extrinsic
scor
Intrinsic
ionelec
ε
ε
ε
ε
ε
+++= 1.2
Each contribution stems from a short-range movement of charges that responds to
an electric field on different time scales and, hence, through a Fourier transform, in
different frequency regimes. If the oscillating masses experience a restoring force, a
relaxation behavior is found (for orientation and space charge polarization). Resonance
effects are observed for the ionic and electronic polarization. The dispersion of the
dielectric function is shown in Figure 1.2, and holds the potential to separate the different
dielectric contributions.
Figure 1.2: Real and imaginary part of the dielectric constant as a function of frequency.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 5
Depending on the local conductivity, the space charge polarization may occur over a
wide frequency range from mHz up to MHz. The polarization due to the orientation of
electric dipoles can take place over a wide frequency range, e.g. from mHz up to a few
GHz in liquids. In the infrared region between 1 and 10 THz, resonances of the molecular
vibrations and ionic lattices constituting the upper frequency limit of the ionic
polarization are observed. The resonance of the electronic polarization is around 1015 Hz.
The dispersion of the dielectric response of each contribution leads to dielectric
losses of the matter which can be mathematically expressed by a complex dielectric
permittivity:
ε
ε
ε
′′
+
′
=i 1.3
Dielectric losses are usually described by the loss tangent:
ε
ε
δ
′
′′
=tan 1.4
It should be taken into account that the general definition of the tan δ is related to the ratio
of loss energy and reactive energy (per period), i. e. all measurements of the loss tangent
also include possible contributions of conductivity σ of a non-ideal dielectric given by
εω
σ
δ
′
=tan 1.5
1.2.3 Crystalline Material Classifications
Depending on their geometry, crystals are classified into seven systems: triclinic
(the least symmetrical), monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and
cubic. These systems can be subdivided into point groups (crystal classes) according to
their symmetry with respect to a point [15]. There are 32 such crystal classes. Out of the
32 feasible crystal classes, 11 have a centre of symmetry and as a consequence cannot
exhibit polar properties (Figure 1.3). The remaining 21 are noncentrosymmetric and
therefore can possess one or more polar axes. Of these, 20 classes are piezoelectric
(excluding the point group 432 being cubic class [16]). Piezoelectric crystals possess the
trait that the application of mechanical stress induces polarization, and equally, the
application of an electric field produces mechanical deformation. Out of the 20
piezoelectric classes, 10 own a unique polar axis and so are spontaneously polarized, i.e.
polarized in the absence of an electric field. Such crystals are called pyroelectric and the
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 6
corresponding spontaneous moment in these crystals can be observed by varying the
temperature.
Figure 1.3: A classification scheme for 32 point groups of crystallography.
Ferroelectric crystals, a restricted group of pyroelectric family, also show the
additional characteristic that the direction of the spontaneous polarization can be inverted
(reoriented) on application of an electric field. Hence, every single ferroelectric is
pyroelectric and piezoelectric but the entire piezoelectrics are not pyroelectric and all
pyroelectrics are not ferroelectric. Relaxor ferroelectrics are a sub group of ferroelectrics.
1.3 Crystal Structures
Among the different crystal structures the perovskite and spinels are more important
with respect to device applications. Most of the studied ferromagnetic, ferroelectric and
relaxor ferroelectric materials belong to these two groups. In particular perovskite oxide
ferroelectrics ABO3 are most thoroughly studied because of their simple structure, and
stability to chemical and physical stimuli. Additionally, substitutions of various kinds of
ions with different ionic radii into A and B sites induce various physical properties, e. g.,
ferroelectricity, anti-ferroelectricity, relaxor, quantum paraelectricity, and multiferroic
natures.
Ch
a
S
tu
d
1
lea
d
tita
n
str
u
this
the
coo
r
sho
w
in t
h
situ
Ca
+
cub
eit
h
Fig
u
in t
h
fold
is p
o
mis
s
lar
g
str
u
this
Pb(
F
a
pter 1
d
ies of Ferro
e
1
.3.1
The maj
o
d
titanate (
P
n
ate (PLZ
T
u
cture.
The basi
c
form at hi
g
larger cati
o
r
dinated a
n
w
s that the
h
ree dime
n
ated in the
c
+
2
etc.) occ
u
e of eigh
t
h
er A
2+
B
4+
O
u
re 1.4: (a)
T
h
e 12-fold ox
y
coordinatio
n
o
ssible. Whe
n
s
ing charge i
s
The per
o
g
e A and
O
u
cture is ve
r
reason it c
F
e
1/2
Ta
1/2
)
O
e
lectric and
M
Perovskit
o
rity of the
P
bTiO
3
), l
e
T
) and mul
t
c
perovskit
e
g
h temper
a
o
n and B
i
n
d the A ca
t
simple cu
b
n
sions with
c
enter of t
h
u
pying the
such octah
O
32-
or A
1+
B
5
T
he typical p
e
y
gen coordi
n
n
). Full or p
a
n
the overall
s
made up b
y
o
vskite stru
c
O
ions with
r
y acceptin
g
a
n be the f
o
O
3
, Pb(Co
1/
4
M
ultiferroic B
e
e Structu
r
valuable
fe
e
ad zircona
t
t
iferroics
R
e
structure
a
ture. In th
e
i
s the sma
l
t
ion is 12-
f
b
ic structur
e
smaller, hi
g
h
e octahedr
a
12-fold co
edra. Most
5
+
O
32-
or A
3
e
rovskite str
u
n
ated holes; t
h
a
rtial substit
u
valence of t
h
y
introducing
c
ture can a
smaller B
g
of cation
o
rerunner t
o
4
Mn
1/4
W
1/2
)
O
e
havior in [B
a
r
e
fe
rroelectric
t
e titanate
(
R
MnO
3
(R
is cubic, a
n
e
unit form
u
l
ler cation.
f
old coordi
n
e
contains c
o
g
hly charg
e
a
and the lo
w
ordination
perovskit
e
3
+
B
3+
O
2-
ty
p
u
cture of AB
h
e B cations
u
tion of the
A
h
e A-site and
vacancies at
lso be con
s
ions filling
replaceme
n
o
more co
m
O
3
, etc. [1,
I
NTRO
D
a
(Zr,Ti)O
3
]
1-
y
s, for insta
n
(
PbZr
x
Ti
1-x
O
= Tb, Ho,
n
d many
m
u
la of pero
v
In this str
u
n
ated with
t
o
rner shari
n
e
d cations
(
w
er charge
d
site (AO
12
)
e
-type ferr
o
p
e formula
[
O
3
. (b) The
A
occupy the
m
A
or B catio
n
B-site catio
n
the oxygen l
a
s
idered as
a
the octah
e
n
t at
b
oth
A
m
plex comp
o
17].
D
UCTION
&
y
:[CoFe
2
O
4
]
y
S
n
ce bariu
m
O
3
), lead l
a
Dy) pos
s
m
ixed
ABO
3
v
skite-type
u
cture, the
t
he oxygen
n
g oxygen
o
(
B: Ti
+4
, Zr
d
, larger ca
t
)
formed i
n
electrics a
r
[
17].
A
cations oc
c
m
uch smaller
n
s with catio
n
n
s (n+m) add
s
a
ttice sites.
a
cubic cl
o
e
dral inters
t
A
and B site
o
unds, suc
h
&
BAC
K
GR
O
S
ystem
m
titanate (
B
a
nthanum
z
s
ess the p
e
3
type oxid
e
oxides AB
O
B cation i
anions. Fi
g
o
ctahedra (
B
+4
, Sn
+4
, T
a
t
ions (A: N
a
n
the middl
r
e compou
n
c
upy the lar
g
octahedral
h
n
s of differe
n
s
up to less t
h
o
se-packed
t
itial positi
o
s of lattice,
h
as (
K
1/2
Bi
1
O
UND
| 7
B
aTiO
3
),
z
irconate
e
rovskite
e
s attain
O
3
, A is
s 6-fold
g
ure 1.4
B
O
6
) set
a
+5
, etc.)
a
+
, Ba
+2
,
e of the
n
ds with
g
er spaces
h
oles (six-
n
t valence
h
an 6, the
array of
o
ns. The
and for
1
/2
)TiO
3
,
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 8
Based on the geometrical packing of atoms, Goldschmidt developed the perovskite
tolerance factor t,
()
OB
OA
rr
rr
t+
+
=2 1.6
where rA, rB, and rO are the respective ionic radii. The tolerance factor allows us to
estimate the degree of distortion. Generally it is an excellent starting point to determine a
given combination of ions will form stable perovskite structures. It is reported that
whether stable perovskite structures may be expected within the limit 0.88 < t < 1.09,
using radii corrected for cations coordination numbers. However, the existence of
ferroelectricity cannot be predicted from tolerance factor arguments alone. General
observations have shown that perovskites with t <1 often have a low symmetry distortion
of the unit cell with ferroelectric rhombohedral or monoclinic phases. In case of t >1,
ferroelectric tetragonal symmetry is commonly observed within the perovskite phase.
1.3.2 Spinel Ferrites
Spinel structure materials are closed packed cubic and have the form AB2O4 where
A represents divalent cations and B trivalent cations. The exchange interaction between A
and B sites is negative and the strongest among the cations so that the net magnetization
results from the difference in magnetic moment between A and B sites. Spinel ferrites or
ferro-spinel are ferromagnetic materials with the general chemical composition
MeO.Fe2O3 where Me is a divalent metal such as iron, manganese, magnesium, nickel,
zinc, cadmium, cobalt, copper, or a combination of these. The Fe ions are the trivalent
variety, Fe3+. The cubic unit cell of spinel structure is illustrated in Figure 1.5.
The spinel structure belongs to space group Fd3m. The cubic unit cell is formed by
56 atoms, 32 oxygen anions distributed in a cubic close packed structure, and 24 cations
occupying 8 of the 64 available tetrahedral sites (A sites) and 16 of the 32 available
octahedral sites (B sites) [18, 19]. The structural formula for a generic spinel compound
MeFe2O4 can be written as [20]
[][]
421 OFeMeFeMe B
ii
A
ii −−
where the amounts in brackets represent the average occupancy of A sites and B sites and
i is the inversion parameter. Depending on cation distribution, a spinel can be normal,
inverse, or partially inverse.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 9
Figure 1.5: (a) Spinel unit cell structure, (b) octahedral interstice (B site: 32 per unit cell, 16
occupied), and (c) tetrahedral interstice (A site: 64 per unit cell, 8 occupied). In (a) the ionic positions
are the same in octants sharing only one edge and different in octants sharing a face. Each octant
contains 4 Oxygen ions. In (a) ionic positions in only two adjacent octants are shown, where the
octant on the left contains octahedral and the one on the right contains tetrahedral sites. All ions are
positioned on body diagonals of the octants and the octant on the right contains a tetrahedral site at
the octant centre.
In a normal spinel structure, the 8 bivalent cations are all located in tetrahedral sites
and the 16 trivalent cations are all located in octahedral sites, while, in an inverse spinel
structure, the 8 bivalent cations occupy 8 octahedral sites and the 16 trivalent cations are
distributed between 8 tetrahedral and 8 octahedral sites [21]. For a normal spinel, і = 0,
and for an inverted spinel, і = 1. If the bivalent cations are present on both tetrahedral and
octahedral sites, the spinel is partially inverted and 0 < і < 1.
1.3.3 Spinel-Perovskite Composites
The intrinsic similarities in crystal chemistry (i.e., oxygen coordination chemistry)
between the perovskite and spinel families lead to compatible lattice dimensions. For
example, the perovskites have a lattice parameter of approx. 4 Å, which is generally
within 5% of the basic building block of the spinels. One can combine these two
structures with different properties to prepare composite materials.
1.4 Ferroelectrics
Ferroelectrics are materials in which spontaneous electrical polarization P
S, can be
reversed by application of an electric field [22, 23]. The nonlinear relationship between the
polarization and the electric field is one of the major and leading features of
ferroelectrics.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 10
Ferroelectrics are dielectric materials that additionally exhibit both the piezoelectric
and the pyroelectric effect. Importantly, ferroelectrics possess at least two energetically
equivalent spontaneous polarization directions, where the local polarization of the crystal
can be reoriented between these states by means of an external electric field. A
macroscopic net non-zero polarization is created by application of a sufficiently high
electric field that gives rise to a preferential alignment of the initially random oriented
individual PS vectors. The most important features of ferroelectrics are briefly presented
in the next section, followed by a description of relaxor ferroelectricity, a phenomenon
closely related yet substantially different from conventional ferroelectricity.
1.4.1 Ferroelectric Phase Transition
The development of spontaneous polarization in a material, often accompanied by
the change of crystal structure is called a ferroelectric phase transitions. The phase
transition occurs due to changes in the forces of interaction between the atoms in the
crystal. An asymmetrical shift in the equilibrium ion positions gives rise to polarization
catastrophe and hence a permanent dipole moment in the material. The spontaneous
polarization (PS) of the ferroelectric system increases with decrease in temperature and
appears discontinuously or some time continuously at a certain temperature, called Curie
temperature or transition temperature (TC). In the ferroelectric phase, at least one set of
ions in the crystal sits in either of the degenerate levels of a double well potential. Above
TC, the particles in the double well have enough kinetic energy to move back and forth
over the barrier that separates the wells, so that the time average position of the atom is
midway between the wells.
a) First and Second Order Phase Transitions
There are two types of phase transitions, first and second order, depending on how
the order parameter changes during the transition [14]. In a ferroelectric system, the
polarization P is the order parameter. The order of the phase transition is defined by the
discontinuity in the partial derivatives of the free energy (Eq. 1.7) of the ferroelectric at
the phase transition temperature.
()
44444443444444421
Landau
F
oPPPEPETPF ⋅⋅⋅⋅⋅+++++−= 6
6
4
4
2
26
1
4
1
2
1
,,
ξξξξ
1.7
where ξn the coefficients depends on the temperature.
Ch
a
S
tu
d
fun
c
per
m
dis
c
tra
n
te
m
Fig
u
ord
e
inc
l
Thi
s
dep
ferr
res
p
Cu
r
stat
e
sec
o
Bri
e
reo
r
a
pter 1
d
ies of Ferro
e
For an n
t
c
tion at t
h
m
ittivity ch
c
ontinuous
a
If there i
s
n
sforms fr
o
m
perature.
u
re 1.6: Free
e
r phase tra
n
Near the
l
uding diel
e
s
is due t
o
endence o
f
oelectric cr
y
C and θ
p
ectively.
T
r
ie point is
e
and true
c
o
nd order
p
e
fly, the C
u
r
ient the sp
o
e
lectric and
M
t
h-order ph
h
e transitio
ange conti
n
a
t for a firs
t
s
more tha
n
o
m one fe
Energy F(
P)
n
sition for E
=
Curie poi
n
e
ctric, elast
i
o
the dist
o
f
the diel
e
y
stals is go
v
in the Eq
.
T
he Curie-
W
the literal
c
onversely.
p
hase trans
i
u
rie-Weiss l
o
ntaneous
p
M
ultiferroic B
e
ase transiti
o
n tempera
t
n
uously at t
h
t
-order ferr
o
n
one ferro
e
rroelectric
P)
, polarizati
o
=
0 [25].
n
t or phas
e
i
c, optical,
a
o
rtion in t
h
e
ctric con
s
v
erned by t
h
(
ε
.
1.8 are c
a
W
eiss temp
e
temperatur
e
For a first
i
tion the t
w
aw is a ch
a
p
olarization
e
havior in [B
a
o
n, the nth
-
t
ure. The
h
e second
o
o
electric ph
a
e
lectric ph
a
phase to
o
n P(T), and
e
transitio
n
a
nd therma
h
e crystal
a
s
tant abov
e
h
e Curie-
W
(
)(
=>
T
TT
C
a
lled Curi
e
e
rature
θ
is
u
e
of the tr
a
order phas
e
w
o are equ
a
a
racteristic
o
is an essen
t
I
NTRO
D
a
(Zr,Ti)O
3
]
1-
y
-
order deri
v
spontaneo
u
o
rder ferroe
l
a
se transiti
o
a
se, the te
m
another a
r
permittivity
n
temperatu
l constants
a
s the pha
e
the Curi
W
eiss law:
)
θ
−
T
C
e
constant
a
u
sually dif
f
a
nsition fro
m
e
transition
a
l (i.e.
T
C
=
o
f ferroelec
t
t
ial require
m
D
UCTION
&
y
:[CoFe
2
O
4
]
y
S
v
ative of
F
u
s polariza
t
l
ectric pha
s
o
ns (Figure
m
peratures
a
r
e also ca
l
ε(T), ε
-1
(T)
f
re, thermo
d
show an a
n
se change
s
e point (
T
a
nd Curie-
W
f
erent from
m
ferroele
c
θ
is less t
h
=
θ
), see
F
t
rics, wher
e
m
ent of fer
r
&
BAC
K
GR
O
S
ystem
F
is a disco
n
t
ion and
d
s
e transitio
n
1.6) [24].
a
t which th
e
l
led the t
r
f
or a first a
n
d
ynamic p
r
n
omalous b
s
. The te
m
T
>T
C
) in
m
W
eiss tem
p
the Curie
p
c
tric to par
a
h
an
T
C
whe
r
F
igure 1.6
[
e
as the cap
a
r
oelectricit
y
O
UND
| 11
n
tinuous
d
ielectric
n
and are
e
crystal
r
ansition
n
d second
r
operties
b
ehavior.
m
perature
m
ost of
1.8
p
erature,
p
oint
T
C
.
a
electric
r
eas in a
[
26, 27].
a
bility to
y
.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 12
1.4.2 Ferroelectric Hysteresis
Ferroelectric materials retain an orientation and magnitude of the polarization even
after the removal of external electric field. Ferroelectric crystals show a hysteresis loop
(Figure 1.7) between polarization (P) and electric field (E)
Figure 1.7: Typical ferroelectric hysteresis loop.
First application of an electric field to a natural ferroelectric sample initiates the
development of domains in the direction of applied field. The polarization reaches its
saturation level (PS) for effectively high electric fields. Ferroelectric materials possess
remnant polarization (Pr) even when the field is reduced to zero. On the reverse direction
of field, first the polarization is reversed to zero and then changes its direction as the field
produces saturation polarization in the opposite direction. The field at which polarization
becomes zero is called the coercive field (
EC). The complete cycle of an alternating
electric field gives rise to a hysteresis loop (Figure 1.7) between polarization and applied
electric field.
1.4.3 Displacive and Order Disorder Type Phase Transitions
Ferroelectric phase transitions are classified into two categories: a displacive type
and order-disorder types. In the displacive ferroelectric phase transition, a macroscopic
polarization occurs as the result of relative displacements of cations and anions below TC.
This type of phase transition is characterized by the development of a soft phonon. With
decreasing temperature, the frequency of a low lying transverse optic mode (TO mode)
decreases toward zero at the Brillouin zone center. At TC where a dielectric constant takes
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 13
a maximum value, it becomes equal to zero and the structural phase transition to a polar
state is induced as a result of the lattice instability. This is caused by the competition
between a long range Coulomb force which favors a ferroelectric phase and a restoring
short range force driving to a paraelectric state. On the contrary, in the order-disorder
type, permanent dipole moments of molecules exist even above TC. Above TC, the
directions of dipole moments are randomly distributed and a net polarization does not
occur. Below TC, they are aligned in a specific direction and a spontaneous polarization
appears. The critical slowing down of the relaxation time which becomes infinite towards
TC, is a characteristic of this transition type.
In reality, both types of ferroelectric phase transitions coexist and it is rather
difficult to distinguish which is dominant, particularly around the phase transition
temperature.
1.4.4 Barium Titanate
Barium Titanate (BaTiO3) is a perovskite structured material, which is ferroelectric,
piezoelectric and pyroelectric. It has been one of the best Pb-free ferroelectric materials
for microelectronic devices applications in perovskite oxide families and also an ideal
model for discussing of the ferroelectricity and spontaneous polarization from the point of
view of crystal structure. At room temperature, BaTiO3 is ferroelectric exhibiting a
spontaneous polarization of 26 μC cm−2 [28]. Using the example of barium titanate, both
the perovskite structure and phase transitions will now be summarized:
The idealized cubic structure can be realized in BaTiO3 above the Curie
temperature. Barium ions Ba2+ (A ions), which are large in size (~1.58 Å), occupy the
corner sites, titanate ions Ti4+ (B ions), which are small in size (~0.605 Å) is at the center
of unit cell, and oxygen anions O2- in the face centers as illustrated by Figure 1.8 [29].
Since the center of both positive and negative charges coincide, BaTiO3 is paraelectric
(PE) at temperatures beyond the transition temperature TC = 393 K [30, 31]. Above the
Curie temperature TC, the cubic structure is stable with a = 4.009 Å [32, 33].
At the Curie temperature, the crystal undergoes a phase transition which is a
displacement type phase transition [2, 32]. Therefore, the crystal structure of BaTiO3 is
also temperature and stress dependent, as shown in Figure 1.8 and Figure 1.9 (a). When
cooling through the Curie temperature between 393 K and 278 K, BaTiO3 gains an
elongation along the c–axis whilst the a and b–axes of the unit cell slightly contract and
Ch
a
S
tu
d
the
(Fi
g
Fig
u
In t
h
Fig
u
latti
and
a
pter 1
d
ies of Ferro
e
tetragonal
s
g
ure 1.8 an
d
u
re 1.8: The
u
h
e cubic pha
s
u
re 1.9: Pro
p
ce dimensio
n
the c directi
o
e
lectric and
M
s
tructure (
a
d
Figure 1.
9
u
nit cells for
t
s
e a = b = c =
p
erties of ba
r
n
s after [29]
b
o
n after [34].
M
ultiferroic B
e
a
= 3.992
Å
9
(a)
).
t
he four pha
s
4.009 Å.
r
ium titanat
e
b
) spontane
o
e
havior in [B
a
Å
,
c
= 4.03
5
s
es of BaTiO
3
e
as a functi
o
o
us polarizat
i
I
NTRO
D
a
(Zr,Ti)O
3
]
1-
y
5
Å) is for
m
3
. Arrows in
d
o
n of the te
m
i
on c) relati
v
D
UCTION
&
y
:[CoFe
2
O
4
]
y
S
m
ed with o
c
d
icate directi
o
m
perature:
a
v
e permittivi
t
&
BAC
K
GR
O
S
ystem
c
tahedron d
i
o
n of the pol
a
a
) unit-cell d
i
t
ies measure
d
O
UND
| 14
i
stortion
a
rization.
i
stortions
d
in the a
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 15
In the tetragonal phase, the polarization points along the 〈001〉 direction, aligning
with the elongated c–axis of the tetragonal BaTiO3 lattice. The electric dipole moment of
the tetragonal BaTiO3 unit cell is caused by a slight displacement of the O2− ions with
respect to the Ba2+ and Ti4+ ions. The small displacements of the ions cause a net electric
dipole, which in turn produces the spontaneous polarization in BaTiO3. At 278 K, BaTiO3
undergoes a second structural phase transition from tetragonal to orthorhombic. As a
result of this phase transition the ferroelectric polarization rotates from 〈001〉 to 〈011〉
(Figure 1.8 and Figure 1.9 (b)) [35]. In the orthorhombic phase the O2− ions are displaced
in the direction of the polarization and are also slightly displaced towards the nearest Ti4+
ion [36]. The Ti4+ is displaced anti-parallel to the polarization and the Ba2+ ions continue
to occupy the corners of the unit cell. The unit cell is slightly elongated in the direction of
the polarization.
The final phase transition from orthorhombic to rhombohedral occurs at 183 K. At
this phase transition the polarization rotates from 〈110〉 to 〈111〉 (a = b = c = 4.004 Å)
with α = 90°. The O2− ions are displaced in the direction of the polarization and Ti4+ ion is
displaced anti-parallel. This off-centering is stabilized by energy-lowering covalent bond
formation, in which charge transfers from the filled oxygen 2p-orbitals into the d-states of
the transition metal ion, which must be empty for this mechanism to be favorable.
The phase transition in barium titanate is of first order (see Figure 1.9), and as a
consequence, there is a discontinuity in the polarization, lattice constant and dielectric
constant as evidenced on Figure 1.9. This first order behaviour is also evidenced by a
small thermal hysteresis of the transition, which depends on many parameters such as
mechanical stress, crystal imperfections, or impurities. From a chemical point of view, the
Ba-O framework evokes an interstice for the central Ti4+ ion which is larger than the
actual size of the Ti4+ ion. As a result, the series of phase transformations takes place to
reduce the Ti cavity size. Certainly, the radii of the ions involved impact the propensity
for forming ferroelectric phases; thus BaTiO3 have ferroelectric phases, while CaTiO3 and
SrTiO3 do not [37].
1.5 Antiferroelectrics
An antiferroelectric crystal is composed of two collinear sublattices with equal and
opposite polarizations. In general, one may have two independent sublattice polarizations
Pi and Pj corresponding to the two sublattices.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 16
In this phenomenological model of antiferroelectricity, the dielectric constant above
the transition temperature is given by
() ()
[]
θλ
ε
−+
=> Tg
TT C
1 1.9
This expression can be recast to the Curie-Weiss form
()
()
θ
ε
−
=> T
C
TT C
by choosing
λ
/1=C, and
λ
θ
/gTC−= . Obviously, for
θ
λ
>/g, TC will be negative.
1.6 Diffuse Phase Transition
Many phase transitions in macroscopic homogeneous materials are characterized by
the fact that the transition temperature is not really sharply defined. In these so called
diffuse phase transitions (DPT), the transition is smeared out over a certain temperature
interval, resulting in a gradual change of physical properties in this temperature range.
This phenomenon is observed in several types of materials, but the most remarkable
examples of DPT are found in ferroelectric materials. Characteristics for the DPT
behaviour in this case are: [38]
a) Broadened maxima in the permittivity curves.
b) Gradual decrease of spontaneous (remanent) polarization with rising
temperature.
c) Transition temperatures as obtained by different techniques do not coincide.
d) Relaxational character of the dielectric properties in the transition region.
e) No Curie-Weiss behaviour in a certain temperature interval above the
transition temperature.
The diffuseness of the phase transition is assumed to be due to the occurrence of
compositional and polarization (structural) fluctuation in a relatively large temperature
interval around the transition. Polarization fluctuation is due to the small energy
difference between high and low temperature phases around the transition. Complex
perovskite type ferroelectrics with distorted cation arrangements show DPT which is
characterized by a broad maximum for the temperature dependence of dielectric constant
(εʹ) and dielectric dispersion in the transition region [39, 40]. For DPT εʹ(T) follows
modified Curie Weiss behavior [41]
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 17
()
21
)(
11
1
≤≤
′
−
+
′
=
′
γ
εε
γ
C
TT m
m
1.10
where Tʹm is the temperature at which εʹ reaches maximum, ɛʹm is the value of εʹ at Tʹm, C1
is the modified Curie Weiss like constant and γ is the critical exponent, which explains
the diffusivity of the materials, which lies in the range 1< γ <2 [42]. The smeared out εʹ
vs. T response has generally been attributed [1, 40, 43] to the presence of nanoregions
with local compositions varying from the average composition over length scale of 100 to
1000 Å. The transition of different microregions in a macroscopic sample are assumed [1]
to be at different temperatures, so-called Curie range, leading to DPT which is due to
compositional fluctuations. The dielectric and mechanical properties of FE system below
their TC are in general functions of the state of polarization and stress.
1.7 Relaxor Ferroelectrics
Chemical or physical disorder e.g. due to substituent ions can lead to a complex
behavior in ferroelectrics. One possible consequence of disorder is the so-called
“relaxor” behavior [1], and it normally results from compositionally induced disorder or
frustration in ferroelectric materials. Relaxor behaviour is associated with a gradual
transition from macroscopic paraelectric to a ferroelectric phase at a temperature below
that of the peak in permittivity.
Extensive research on relaxor ferroelectrics (RFE) has been carried out since 1950s.
There are several groups in the world who have defined the evolution and origin of
relaxor properties in the ferroelectric materials in their own way. There are two common
points of consensuses among the scientists. One is the presence of polar nano regions
(PNRs). These are separate regions of the crystal which have nanometer scale size and
possess spontaneous polarization. PNRs are self-assembled domains of short range
ordering typically of less than 20-50 nm in the ferroelectric relaxor materials which
causes the dielectric dispersion near the phase transition temperature. Second agreement
is over the existence of random dipolar fields at the nanoscale.
Relaxor ferroelectrics have been divided into two main categories,
a) Classical relaxors (only short range ordering), the most common example is
PbMg1/3Nb2/3O3 (PMN).
b) Semi-classical relaxor ferroelectrics (a combination of short and long range
ordering).
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 18
In the latter case, the relaxor properties can arise due to compositional
inhomogeneities, artificially induced strain, growth conditions (temperature, pressure,
medium, etc.), and due to different ionic radii mismatched based chemical pressure in the
matrix. The local domains (PNRs) reorientation induces polar-strain coupling which
makes RFE potentially high piezoelectric coefficient materials widely used in
Micro/Nano electromechanical system (MEMS/NEMS). The basic features of the RFE
over a wide range of temperatures and frequencies are as follows
a) Dispersive and diffuse phase transition,
b) Partially disordered structure,
c) Existence of polar nano-ordered regions, etc. over a wide range of
temperatures and frequencies.
These effect was first observed by Smolenskii and Isupov (1954) in the solid
solution of Ba(Ti1-xSnx)O3 [44].
1.7.1 Macroscopic Properties of Relaxor Ferroelectrics
The macroscopic properties of relaxor ferroelectrics are quantitatively different
from proper ferroelectrics. Relaxor ferroelectrics deviate from ordinary ferroelectrics in
terms of their frequency and temperature dependent dielectric response. In order to
appreciate and understand the properties of relaxors, it is useful to compare some of their
properties with those of normal ferroelectric. We can do so with the help of Figure 1.10
and Table 1.1.
In contrast to the rectangular polarization hysteresis observed for conventional FE,
the P(E) loops of RFEs may remain slim with little remanence and minute coercive field.
Moreover, the temperature-driven PE-FE phase transition of ferroelectrics is associated
with symmetry breaking and a peak in permittivity at TC, above which the Curie-Weiss
law applies and PS = 0. While the εʹ(T) curve of RFEs also exhibits a maximum at
temperature Tm, the values of both Tm and εʹ(T=Tm) depend on the measurement
frequency. This effect, referred to as frequency dispersion, is one of the most
characteristic features of RFEs. In contrast to FE, this peak in permittivity is not
associated with a structural phase transition and the permittivity does not obey the Curie-
Weiss law for all T >Tm. Also, the remanent polarization may assume finite nonzero
values in excess of Tm.
Ch
a
S
tu
d
Fig
u
pol
a
(aft
e
Tab
l
P
r
Te
m
D
i
e
Fr
e
D
i
e
εʹ
(
P
a
Re
S
c
a
X-
r
b
ut
cap
a
pos
i
shu
t
a
pter 1
d
ies of Ferro
e
u
re 1.10: C
o
a
rization (le
ft
e
r Ref. [45])
l
e 1.1: Comp
r
opert
y
m
perature
D
e
lectric Co
n
e
quenc
y
De
e
lectric Co
n
(
T) Behavi
o
raelectric
R
e
manent Po
a
ttering of
L
r
ay Diffra
c
Relaxor
f
also due
a
citors, hi
g
i
tioning, a
n
t
ters or opt
i
e
lectric and
M
o
mparison o
ft
), temperat
u
arison betw
e
D
ependen
c
n
stant εʹ(T
)
e
pendence
o
n
stant εʹ(f)
o
r in
R
egime
larization
L
igh
t
c
tion
f
erroelectri
c
to their hi
g
h electro
s
n
d the ele
c
i
cal modula
t
M
ultiferroic B
e
f ferroelect
r
u
re-depende
n
e
en normal a
n
No
r
c
e of
)
Shar
p
Tran
Tem
p
o
f Wea
k
Obe
y
=
′
ε
1
Larg
Stro
n
(Bir
e
Line
stru
c
c
s, howeve
r
gh applica
b
s
trictive c
o
c
tro-optical
t
ors.
e
havior in [B
a
r
ics and rel
a
n
t polarizati
o
n
d relaxor fe
r
r
mal Ferr
p
1
st
& 2
nd
o
sition abou
t
p
erature
T
C
k
y
s Curie-W
e
C
TT
CW
−
e
n
g Anisotro
e
fringence)
splitting b
e
c
tural phase
r
, do not o
n
b
ility. Lar
g
o
efficients
characteri
I
NTRO
D
a
(Zr,Ti)O
3
]
1-
y
a
xor ferroel
e
o
n (middle),
r
roelectrics [
oelectric
o
rder
t
Curie
e
iss Law
py
e
low the
transition
n
ly stand o
u
g
e dielectri
are explo
i
stics are
e
D
UCTION
&
y
:[CoFe
2
O
4
]
y
S
e
ctrics in te
and relativ
46].
Relaxo
r
Diffuse
T
Curie M
a
Strong
Obeys
M
Weiss L
a
11
m
εε
′
=
′
Small
Very W
e
(Pseudo
C
No X-ra
y
to pseud
o
u
t because
c constant
s
i
ted in ac
t
e
mployed i
n
&
BAC
K
GR
O
S
ystem
rms of fiel
d
e permittivi
t
r
Ferroel
e
T
ransition a
b
a
ximum
Tʹ
m
M
odified Cu
r
a
w
1
)(
C
TT
m
γ
′
−
+
e
ak Anisotr
o
C
ubic)
y
line splitti
n
o
cubic stru
c
of these p
r
s
are bene
f
t
uators an
d
n
devices
O
UND
| 19
d
-induced
t
y (right)
e
ctric
b
out
m
r
ie-
γ
o
py
n
g due
c
ture
r
operties
f
icial in
d
micro
such as
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 20
1.8 Theoretical Models
Over past sixty years, number of efforts has been made to model the dielectric
response of the relaxor ferroelectric using various hypotheses concerning the origin and
behavior of the polar nano regions. Some of these models are: statistical composition
fluctuations model [40, 44] superparaelectric model [1], dipolar glass model [47, 48],
random field model [49-52], breathing model [53], random-bond–random-field model
[54], spherical random field model [55] etc. These models can explain much of the
experimental observed facts but a clear understanding of the relaxor nature or even a
comprehensive theory is not available yet. Despite the absence of a comprehensive theory
of relaxor ferroelectricity, literature agrees to define relaxor materials in terms of the
existence of polar nano regions as mentioned above, these ordered regions exist in a
disordered environment. Here we review some of the above empirical and theoretical
models which are used to explain the dielectric behavior of the relaxors. A comprehensive
overview on the models is provided by Bokov and Ye [4].
1.8.1 Compositional Fluctuation Model
The first attempt to model dielectric behavior of the relaxor was made by
Smolenskii and Isupov [40, 44]. They introduced the concept of the “diffuse phase
transition”. They explain the diffuseness in the transition temperature by chemical
heterogeneity on the microscopic scale due to composition fluctuations on a microscopic
scale. According to this model, the phase transition from non-polar to polar state occurs
locally in separate regions of the crystal, with size 10-100 nm. These regions are
independent from one and another, with the local transition temperature, TC, loc depending
upon the composition of individual region. These regions are large enough to allow the
occurrence of spontaneous polarization PS and are called the polar nano regions (PNR).
The dielectric response was interpreted as a switching of the local spontaneous
polarization between states with different orientations of the PS. At a given temperature T,
only regions of which the local TC, loc is close to T contribute to the macroscopic dielectric
response.
Burns and Dacol [56] showed that local polarizations or polar nanoregions (PNRs)
appear at a dipole temperature Td (also referred to as the Burns temperature TB), which
may be several hundred Kelvins above Tm. The presence of PNRs between TB and Tm was
also supported by the non-linear behavior of the thermal strain and the thermal expansion
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 21
coefficient attributed to the electrostrictive effects which have not only been observed in
disordered perovskite but also in highly ordered perovskites like PbSc1/2Ti1/2O3 (PST).
Also the deviation from the linear Curie-Weiss law suggests the presence of PNRs above
Tm.
1.8.2 Superparaelectric Model
Superparaelectric model was proposed by the L. E. Cross in analogy with the
superparamagnetic materials [1]. He suggested that the relaxor might be considered as
superparaelectrics. He visualized the relaxors as consisting of small non-interacting polar
nanoregions each with a local spontaneous polarization. Cross postulated that since
ferroelectricity is a cooperative process, the energy involved with every polar region must
scale with volume and that for regions with a size of ~10 nm. The potential barrier
required to reorient the local polarization vector would be comparable to the thermal
energy of the crystal. Thus, the local polarization in each region can fluctuate under the
thermal agitation. The permittivity relaxation in relaxor ferroelectrics is due to thermally
activated polarization reversals between the equivalent polar variants.
The frequency of the fluctuations decreases with decreasing temperature leading the
observed dispersion in the permittivity response. Eventually at the low temperature, for
low thermal energies and at finite frequencies, a subset of the relaxation times become too
slow to respond to the applied measurement field and effectively become “frozen” into
one of the polarization states. The superparaelectric model successfully explains many of
the features associated with relaxor ferroelectric behavior.
1.8.3 Dipolar Glass Model
Viehland et al. [47, 48], proposed dipolar glass model to include the cluster
interactions in analogy with magnetic spin glasses and related dipolar glasses. According
to this model, individual clusters interact with each other by means of dipolar
interactions. As the temperature decreases, the strength of the interactions increases and at
sufficiently low temperature Tf, the originally dynamically disordered polar regions will
“freeze” into metastable or frustrated polar states such that clusters may not change their
polarization anymore due to the presence of the random fields produced by the structural
disorder.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 22
1.8.4 Vogel-Fulcher Law
The dielectric relaxation in relaxor ferroelectrics does not fit the classical Debye
relaxation model; instead there is a distribution of relaxation times related to the sizes of
the nano domains. The dynamics of polar nanoregions does not follow Arrhenius type
temperature dependence; instead nice fit of the frequency dispersion for each relaxor
system is obtained with the Vogel-Fulcher law [57, 58] (originally derived for the
magnetic spin-glass systems) is described as follows:
()
⎥
⎦
⎤
⎢
⎣
⎡
−
−=
VFmB
a
TTk
E
ff exp
ο
1.11
where fₒ is the attempt frequency which is related to the cut-off frequency of the
distribution of relaxation times, Ea is the activation barrier to dipole reorientation, Tm is
the dielectric maxima temperature, and TVF is the freezing temperature where polarization
fluctuations “frozen-in”. Later, Pirc and Blinc [59] derived the same relation for the
relaxor ferroelectrics by introducing a mesoscopic mechanism for the growth of PNRs.
1.8.5 Random Field Model
The quenched-random-field approach is based on the theoretical work of Imry and
Ma [60] proposed by Westphal, Kleemann and Glinchuck [50, 61]. They suggest that the
existence of stabilized size-restricted nano-regions is due to the interplay between the
surface energy of domain walls and the bulk energy of domains in the presence of
arbitrary weak random fields induced by compositional fluctuations. In this model the
crystal is represented by low-symmetry nano-domains separated by domain walls with a
thickness approaching the linear size of the nano-domains. In such systems with a
continuous change of the order parameter, a second-order phase transition should be
suppressed by quenched random local fields conjugated to the order parameter. Below the
Curie temperature the system breaks into small-size domains, similar to the concept of
PNRs, instead of forming a long-range ordered state.
Glinchuck & Farhi (1996) proposed a random field theory based on electrostatic
interactions where the transition is regarded as an order-disorder phase transition. Thus at
high temperatures the crystal is represented by a system of random reorientable dipoles
embedded in a highly polarizable matrix similar to the concept of the dipolar glass model.
In contrast to the dipolar glass model, the dipole-dipole interactions are not direct but
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 23
indirect via the matrix leading to uniformly directed local fields and thus to ferroelectric
ordering at low temperatures.
1.8.6 Spherical Random Bond Random Field Model
A spherical cluster glass model or spherical random-bond-random-field (SRBRF)
model involving both random bonds and quenched random fields has been recently
proposed [55]. In this model, relaxor ferroelectric is regarded as an intermediate state
between dipole glasses and normal ferroelectrics. In contrast to dipolar glasses, where
elementary dipolar moments exist on the atomic scale, the relaxor state is characterized
by the presence of nanoscale polar cluster of variable sizes. This picture constitutes the
basis of the superparaelectric model [62] and of the more recent reorientable polar cluster
model of relaxor [63]. The long range frustrated inter cluster interaction of a spin-glass
type is also taken into account into this picture. The system can be described by the
pseudospin Hamiltonian:
∑
∑
−−=
i
ii
ij
jiij PhPPjH 1.12
Here the first sum is the interaction between the dipole moments (pseudospin) P at
lattice site i and j that are coupled by interaction constants Jij with Gaussian distribution.
The second sum denotes the interaction of the dipole moments Pi (pseudospin) with
quenched random field hi, where
∑
=
i
i
h0 , but
∑
≠
i
i
h0
2.
The electric dipoles randomly distributed in the system were treated as the main
sources of random fields. This model is capable of elucidating the static behavior of
relaxors, such as the line shape of quadrapole nuclear magnetic resonance (NMR) in
PMN [64] and PST [65], and the sharp increase of the quasi-static third-order nonlinear
dielectric constant [66]. This model has been extended to describe the dynamic of relaxor
ferroelectrics by introducing Langevin-type equation of motion [67].
1.8.7 Edward Anderson Order Parameter
Relaxors show many properties similar to those of spin and dipolar glasses.
Sherrington and Kirkpatrick [68] developed an infinite range model for spin glass which
linked the temperature dependence of the susceptibility below Tg (dynamic freezing
temperature) to the outset of a local (spin glass) order parameter as
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 24
()
()
EA
EA
qT
qC
−−
−
=1
1
θ
χ
1.13
Where EA
q is the Edward-Anderson (EA) spin glass local order parameter and χ is the
susceptibility (in magnetic system).
Calculation by Binder [69, 70] using Ising model with Gaussian distribution of
correlation lengths between neighboring moments has indicated that EA
q does not go to
zero at Tg but tails to zero at much higher temperatures suggestive of Curie Weiss
deviation about Tg. The divergence from Curie-Weiss behavior in spin glasses has been
explained in terms of strong magnetic correlations developing far above Tg. Similarly it
has been recently shown [71] that the deviation from Curie-Weiss behavior in relaxor
ferroelectrics occurs due to correlations between polar regions (superparaelectric). At Tdev
local ferroelectric transitions are started; as the temperature further decreases, more
correlations are established and hence the volume fraction of polar regions enlarges and
disordering due to thermal agitation decreases. At a lower subsequent temperature Tf, the
polarization fluctuations start to freeze. Such a glasslike freezing of cluster dynamic could
be characterized by the Edward-Anderson spin glass order parameter, EA
q(T).
a) Reconstruction of Edward-Anderson Order Parameter for Relaxors
A necessary condition of relaxor behavior is the simultaneous existence of the broad
distribution of local field and the broad distribution of dipole relaxation frequencies [72].
B. E. Vugmeister and H. Rabitz [63] analyzed the dielectric response of the relaxor
ferroelectric in the framework of a model for polarization dynamics in the presence of the
polar clusters. This theory explicitly includes the distribution of the cluster orientation
frequencies and the effect of cluster-cluster interactions in highly polarizable crystals, and
described these effects in terms of local field distribution function.
B. E. Vugmeister and H. Rabitz performed a reconstruction for the order parameter
of nonequilibrium spin glass state taking into account the competition between local glass
like freezing and the critical slowing down of the dynamics in highly polarizable
materials. In this model they described the dynamical behavior of relaxor ferroelectric
using the Bloch type equation.
()
)(
1EPP
t
Peq
clcl
cl −=
∂
∂
τ
1.14
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 25
Equation 1.14 describes the relaxation of polarization of each cluster to its quasi-
equilibrium value )(EP eq
cl . They further assume that the distribution function f(E,P) of the
local field, which depends parametrically on the average polarization of the system P(t),
has the form
()
⎟
⎠
⎞
⎜
⎝
⎛−−=
π
εγ
γ
4
,exo E
PEfPEf
This form of f(E,P) is consistent with the mean field approximation.
The value γεₒEex/4π is the local field induced by the external field at the location of
each off-center ion in a dielectric media with the dielectric constant εₒ>>1; i.e. we
assume that the localized dipole moments are distributed in crystals with large lattice
permittivity caused by the existence of the soft modes.
In order to reproduce the effect of a non-exponential relaxation within the proposed
formalism Eq. 1.14 is written in the integral form and taking the average with respect to τ,
E, and the initial cluster polarization Pcl(0), one obtains
⎥
⎦
⎤
⎢
⎣
⎡′
−−
′
−
′
′
′
−= ∫)(
4
)(
)(
)()()0()(
0
ttEttP
td
tdq
tdTktqPtP ex
o
t
EA
EA
π
ε
1.15
In Eq. 1.15 we assumed a linear response of the polarization to the applied electric
field, and T >TC where TC is the temperature of a possible ferroelectric phase transition,
()
∫
=
∂
=
t
P
eq
cl dP
PEf
EPdETk
00
),(
)( 1.16
and for second order transitions k→1 as T→ TC : The function EA
q(t) is
∫∫ ≈
⎟
⎠
⎞
⎜
⎝
⎛−=
m
o
m
o
gd
t
gdtqEA
τ
τ
τ
τ
ττ
τ
ττ
)(exp)()( 1.17
where g(τ) is a distribution function of relaxation times and the right-hand side expression
of Eq. 1.17 is valid for smooth functions g(τ). The variable EA
q(t) describes the fraction
of clusters effectively frozen at time t and, therefore, has the meaning of EA spin glass
order parameter on a finite time scale. Note that as we assumed above, EA
q(t)→0 at t→∞:
The steady state frequency dependent permittivity can be easily obtained from Eq.
1.15 assuming
()
tiEtE exex
ω
exp)( )1(
= 1.18
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 26
and using the definition
()
o
ex
dE
P
T
ε
ω
πωε
+
∂
=)1(
)(
4, 1.19
Thus we obtain
()
),()(1
,TQTk
To
ω
ε
ωε
−
= 1.20
Where
()
∫
=
m
odt
dq
tidTQ EA
τ
τ
ωτω
exp),( 1.21
For the smooth function g(τ) the real part of Q(ω, T) can be simplified as [73]
()
Tqgd
i
g
dtTQ EA
o
m
o
,1)(
1
)(
),( 1
/1
−
−=≈
+
=
′∫∫
ωττ
ωτ
τ
ω
ω
τ
τ
τ
1.22
Eq. 1.22 can be employed to obtain information on EA order parameter in relaxors
from frequency dependent dielectric constant using the fact that Qʺ<Qʹ. Nonequilibrium
spin glass order parameter EA
q(t, T) is a very important quantity for determining the
dielectric response of relaxors. The dielectric response can be formulated in terms of the
parameter EA
q(t, T) in a way consistent with the description of magnetic susceptibility in
magnetic alloys where the spin-glass-state and the magnetic order coexist. We can rewrite
Eq. 1.20 in the identical form
()
()
o
EA
EAo
qk
qk
T
ε
ε
ωε
+
−−
−
=
′11
1
),( 1.23
Where EA
q= EA
q(
ω
,T) and k(T) is a parameter whose deviation from T
θ
indicates the
deviation from the mean field picture. The first term of this equation is similar to Eq.
1.13, where T
Tk
θ
=)( . The EA
q in the relaxors is a frequency dependent quantity.
Solving for EA
q the above equation
()
C
CTT
TqEA +
′
+−
′
=
θε
θ
ω
ε
ω
),(
),( 1.24
The reviewed models are well apt for explaining nonlinear dielectric response of
relaxor ferroelectrics. In summary, inhomogeneities cause a complex energy landscape
where polar nano regions must exist.
Ch
a
S
tu
d
1.9
b
y
a
sim
tra
n
ran
d
occ
u
tra
n
ma
c
dra
m
at
T
clo
s
can
ion
s
te
m
clu
s
a
pter 1
d
ies of Ferro
e
Tempe
r
Under th
e
a
sequence
o
At high
t
ila
r
, in ma
n
n
sfor
m
into
d
omly dist
r
u
rs at the
s
n
sition bec
a
c
roscopic o
r
Neverthe
m
atically,
g
T
<T
B
is oft
e
s
e to
T
B
the
be associa
t
s
without l
o
m
perature d
e
s
ters and sl
e
lectric and
M
r
ature E
v
e
condition
o
f characte
r
Figure 1.1
1
t
emperatur
e
n
y respects
the ergodi
c
r
ibuted dir
e
s
o-called B
u
a
use it is
n
r
mesoscop
i
l
ess, the p
g
iving rise t
e
n consider
e
PNRs are
t
ed with th
e
o
cal strain
f
e
crease the
ow down
t
M
ultiferroic B
e
v
olution
o
of temper
a
r
istic temp
e
1
: Evolution
o
e
RFEs exi
, to the P
E
c
relaxor s
t
e
ctions of
d
u
rns temp
e
n
ot accom
p
i
c scale.
olar nanor
e
o unique p
h
e
d as the n
e
mobile an
d
e
formatio
n
f
ields resul
t
PNRs cou
p
t
heir flippi
n
e
havior in [B
a
o
f Polar
N
a
ture decre
a
e
ratures (Fi
g
o
f PNRs upo
n
st in a no
n
E
phase of
n
t
ate in whi
c
d
ipole mo
m
e
rature (
T
B
)
p
anied by
e
gions (P
N
h
ysical pro
p
e
w phase d
i
d
their beha
v
n
of short l
i
t
ing in the
p
le, due to
n
g dynamic
I
NTRO
D
a
(Zr,Ti)O
3
]
1-
y
N
ano Re
gi
a
se the evol
u
g
ure 1.11).
n
temperatu
r
n
-polar par
a
n
ormal fer
r
c
h polar re
g
m
ents appe
a
cannot
b
e
any chang
e
N
Rs) affect
p
erties. For
i
fferent fro
m
v
iour is er
g
i
ved correl
a
formation
their elect
r
s [75-78].
T
D
UCTION
&
y
:[CoFe
2
O
4
]
y
S
i
ons
u
tion of th
e
r
e decrease [
7
a
electric (P
E
r
oelectrics.
g
ions of na
n
a
r. This tr
a
considere
d
e
of cryst
a
the behav
i
this reaso
n
m
the PE o
n
g
odic. It w
a
a
tion betwe
of dynami
c
r
ic field at
T
* can be
&
BAC
K
GR
O
S
ystem
e
PNRs can
7
4].
E
) phase,
w
Upon cool
i
n
ometer sc
a
nsformatio
n
d
a structur
a
a
l structure
i
our of th
e
n
the state o
n
e. At tem
p
a
s suggeste
d
en the off-
c
c
PNRs. O
n
T*, to for
m
related to
t
O
UND
| 27
be seen
w
hich is
i
ng they
ale with
n
which
a
l phase
on the
e
crystal
f crystal
p
eratures
d
that
T
B
c
entered
n
further
m
larger
t
he long
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 28
lived correlation between the off centered ions, which are accompanied with local strain
fields, resulting in the formation of static PNRs. On cooling, their dynamics slows down
enormously and at a low enough temperature, TVF (typically few tens to hundreds degrees
below TB), the PNRs in the canonical relaxors become frozen into a non-ergodic state,
while the average symmetry of the crystal still remains cubic. Similar kind of non-
ergodicity is characteristic of a dipole glass (or spin glass) phase.
The existence of an equilibrium phase transition into a low temperature glassy phase
is one of the most interesting hypotheses which have been intensively discussed. Freezing
of the dipole dynamics is associated with a large and wide peak in the temperature
dependence of the dielectric constant (ε) with characteristic dispersion observed at all
frequencies practically available for dielectric measurements. This peak is of the same
order of magnitude as the peaks at the Curie point in the ordinary ferroelectric (FE)
perovskites, but in contrast to ordinary ferroelectrics it is highly diffuse and its
temperature Tm (>TVF) shifts with frequency due to the dielectric dispersion. Because of
the diffuseness of the dielectric anomaly and the anomalies in the temperature
dependences of some other properties, relaxors are often called (especially in early
literature) the “ferroelectrics with diffuse phase transition” even though no transition into
FE phase really occurs. The non-ergodic relaxor state existing below TVF can be
irreversibly transformed into a FE state by a strong enough external electric field. This is
an important characteristic of relaxors which distinguishes them from typical dipole
glasses. Upon heating the FE phase transforms to the ergodic relaxor one at the
temperature TC which is very close to TVF. In this state, the polar order remains short-
range. Polarization is correlated on the nanometer scale inside the PNRs, but the system
remains nonpolar on a macroscopic scale. As relaxors display various characteristics of
non-ergodic behavior below TVF, viz. an anomalously wide relaxation time spectrum [79]
and dependence of the state on the thermal and field history of the sample [50, 80, 81],
this state is often called the non-ergodic relaxor state [4].
A long-range-ordered ferroelectric-like state may be induced in relaxor compounds
when poling by an electric field larger than a critical strength [82], by applying a
mechanical strain [83], or even by thermal annealing stimulating cation ordering [84].
In many relaxors the spontaneous (i.e. without the application of an electric field)
phase transition from the ER into a low-temperature FE phase still occurs at TC and thus
the non-ergodic relaxor state does not exist.
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 29
1.10 BaTiO3 Based Relaxor System
Barium titanate is one of the most widely used ferroelectrics for capacitor
applications. However, it strongly suffers from temperature dependent dielectric constant
as a result from its sharp phase transitions. These sharp phase transitions and the
existence of the ferroelectric phase at room temperature limits the usefulness of the pure
material in practical applications.
The compositions of most dielectric materials used for ceramic capacitors are based
on ferroelectric barium titanate. As discussed before, the permittivity of ferroelectric
perovskites shows marked changes with temperature, particularly close to the phase
transition. From the device point of view a high dielectric permittivity with stable
properties over a wide temperature range is required. By substitution or doping, it
becomes possible to tailor the ferroelectric materials to different properties. If an ion of
the perovskite structure ABO3 is replaced by a different ion of the same valence, the
doping is isovalent, while heterovalent doping involves doping by different ions of
different valance.
Figure 1.12: The cubic perovskite lattice showing the location of some substituents and vacancies
[45].
A crossover to the relaxor state was observed in ABO3 (BaTiO3) by both
heterovalent and isovalent ionic substitutions. In the case of heterovalent substitutions,
the relaxor behavior is induced by substitutions either on A- or B- or on both A- and B-
sites of the perovskite lattice ABO3, for example, in Ba1-x(Sm0.5Na0.5)xTiO3 [85],
Ch
a
S
tu
d
Ba
1
B
a
T
exa
m
(
Ba
Hf
+
4
te
m
Ba
T
Fi
te
m
wit
h
sta
b
ran
g
do
p
the
the
con
a
pter 1
d
ies of Ferro
e
-x/2
x/2
(Ti
1-
x
T
iO
3
–La(M
g
m
ple on t
h
1−x
Pb
x
TiO
3
)
4
(
BaHf
x
Ti
1
m
perature.
T
T
iO
3
(BTO)
gure 1.13: E
f
A-site d
o
m
perature o
f
h
out any
s
b
ilizes the
t
g
e whereas
p
ing is alw
a
substitutio
n
proper io
n
centrations
e
lectric and
M
x
Nb
x
)O
3
(
g
0.5
Ti
0.5
)O
3
h
e A
2+
-sit
e
)
, and on t
h
1
−x
O
3
) [92-
9
T
he effect
o
is shown i
n
f
fect of sever
a
o
ping catio
n
f
BTO to
e
s
ignificant
t
etragonal
p
S
r
-doping
a
ys associa
t
n
at B-site
o
n
ic size
b
having a st
r
M
ultiferroic B
e
denotes a
v
[89, 90] re
s
e
such as
h
e B
4+
-site
S
9
4]. Dopin
g
o
f various
i
n
Figure 1.
1
a
l isovalent s
u
n
s usually
e
ither decr
e
broadenin
g
p
hase in th
e
destabiliz
e
t
ed with th
e
o
ften result
s
b
ut unmatc
h
r
ong effect
e
havior in [B
a
v
acant site)
s
pectively.
Sr
+2
(
Ba
1−
x
S
n
+4
(
BaSn
x
g
generall
y
i
sovalent d
1
3 [26].
u
bstitutions
o
have the s
e
ase (Sr s
u
g
of phase
e
sense of
e
d the tetr
a
e
off-cente
r
s
in a broa
d
h
ed valen
c
on the diel
e
I
NTRO
D
a
(Zr,Ti)O
3
]
1-
y
[86], and
B
In the seco
n
x
Sr
x
TiO
3
)
o
x
Ti
1−x
O
3
) [
9
y
leads to
a
opants on
o
n the transi
t
ame valen
c
u
bstitution)
transition.
their exist
e
a
gonal pha
s
r
displacem
e
d
ening of p
h
c
e can als
o
e
ctric prop
e
D
UCTION
&
y
:[CoFe
2
O
4
]
y
S
B
aTiO
3
–
B
iS
c
n
d case (s
e
o
r Ca
+2
(
Ba
9
1] or Zr
+4
a
shift of t
h
the three
p
t
ion tempera
t
c
e and cou
or increas
e
It is fou
n
e
nce over
a
s
e. Howev
e
e
nt of Ti
4+
h
ase transit
i
o
enter th
e
rties.
&
BAC
K
GR
O
S
ystem
c
O
3
[11, 8
7
e
e Figure 1.
a
1−x
Ca
x
Ti
O
(
BaZr
x
Ti
1−
x
h
e phase t
r
p
hase trans
i
t
ure of BaTi
O
ld cause t
h
e
(Pb subs
t
n
d that P
b
a
wide te
mp
e
r, since th
in TiO
6
o
c
i
on at
T
C
. I
o
e lattice i
n
O
UND
| 30
7
, 88], or
12). For
O
3
), Pb
+2
x
O
3
) [9],
r
ansition
i
tions of
O
3
[26].
h
e Curie
t
itution),
b
-doping
mp
erature
e B-site
c
tahedra,
o
ns with
n
small
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 31
1.10.1 BaZrxTi1-xO3 Solid Solution
BaZrxTi1-xO3 or (1-x)BaTiO3-xBaZrO3 (BZTx) is most common solid solution for
dielectric applications with zirconium substitutes for titanium in barium titanate
perovskite structure. This is possible because the Zr4+ ion has larger ionic size (0.72 Å)
than the Ti4+ ion (0.605 Å) [95, 96]. This changes some of the properties of BZT. For
example, it becomes a phase transition pincher rather than a shifter (Figure 1.14). It has
been reported that with the incorporation of Zr4+ in BaTiO3, the rhombohedral to
orthorhombic (T1) and orthorhombic to tetragonal (T2) phase transition temperature
corresponding to pure BaTiO3 increase. However, the tetragonal to cubic (TC) phase
transition temperature decreases. Thus when Zr concentration is at about 15 at.%, the
BZT system exhibits a pinched phase transition, and all three phase transition
temperatures (T1, T2 and TC) correspond to pure BaTiO3 are merged or pinched into
single phase transition [9].
1.10.2 Phase Diagram of BaZrxTi1-xO3
Figure 1.14 shows a phase diagram for BaZrxTi1-xO3 solid solution based on
experimental data of dielectric study by several authors [9, 97].
Figure 1.14: Phase diagram of BaZrxTi1-xO3 bulk ceramic [9].
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 32
It is clear from the phase diagram that with the increase in amount of Zr+4
concentration, proper ferroelectric BaTiO3 (x = 0) transforms into a pinched phase
transition at x ~ 0.15 [98]. The pinching effect of the phase transitions, results in a
broadening of the dielectric maximum that can flatten the overall temperature response of
permittivity. At approximately 15 at.% Zr, all phase transitions have converged to a
critical point and the material can transit from cubic directly to rhombohedral [9, 99].
In the composition range 0.15 < x < 0.25, the system depicts almost second order
ferroelectric like diffuse phase transition behavior [9]. The region showing relaxor
behavior in BaZrxTi1-xO3 system has been reported in the composition range 0.25 < x <
0.75 [9]. Phase diagram show polar cluster like behavior for x ≥ 0.80 from a simple
dielectric i.e. pure BaZrO3 (x =1.00) [100].
Barium Zirconate BaZrO3 has the simple perovskite structure with relatively large
lattice constant, higher melting point, smaller thermal expansion coefficient, lower
thermal conductivity, lower dielectric constant, and loss than that of BaTiO3. Because of
these refractory properties BaZrO3 is considered to be a very good candidate to be used as
an inert crucible in crystal growth techniques and an excellent material for wireless
communications etc. [101-105]. BaZrO3 does not show any phase transition up to 1375 K
[106]. There is no strong evidence of the existence of any ferroelectric phase transition in
this compound. Recently Akbarzadeh et al investigated a combined theoretical and
experimental study on BaZrO3 [107].
In summary Barium Zirconate Titanate (BaZrxTi1-xO3) system depending on the
composition, in succession describes the properties extending from simple dielectric (pure
BaZrO3) to polar cluster dielectric, relaxor ferroelectric, second order like diffuse phase
transition, ferroelectric with pinched phase transitions and then to a proper ferroelectric
(pure BaTiO3). Up till now there has been no other single solid solution system that
exhibits such a complex phase diagram.
1.11 Multiferroics
Multifunctional or smart materials combining several properties in the same
structure in order to produce new or enhanced phenomena have stimulated much
scientific and technological interest within the scientific community in the recent years. A
ferroic system has an order parameter switchable by an adequate driving force or field
(phenomena normally accompanied by hysteresis). Ferroelectrics, ferromagnetics and
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 33
ferroelastics are typical examples of ferroics. The electric polarization, magnetic moment
and strain can be switched from one ordered state to the opposite one by means of an
electric, magnetic or stress field, respectively. If the same system, possess two or more
ferroic properties in the same phase, is named multiferroic [108]. Multiferroics with
multiple (charge, spin) order parameters; offer an exciting way of coupling between
electronic and magnetic ordering. In these materials, a weak magnetoelectric interaction
can lead to spectacular cross-coupling effects when it induces electric polarization in a
magnetically ordered state.
Figure 1.15 shows the well-known triangle used to describe the pathways between
external forces, such as stress (σ), electric field (E) and magnetic field (H), and associated
material properties, such as strain (ε), electrical polarization (P), and magnetization (M).
Figure 1.15: Direct interactions between stress (σ) and strain (ε), electric field (E) and polarization
(P), and magnetic field (H) and magnetization (M), are illustrated with the red, yellow, and blue
arrows, respectively. In a single phase multiferroic magnetoelectric material (green arrows), electric
field is directly coupled to magnetic field. In many multiferroic magnetoelectric devices, strain-
coupling (black arrows) between magnetostrictive and piezoelectric phases provides the
magnetoelectric effect [109].
The ferroelastic, ferroelectric, or magnetic ferroic materials exhibiting spontaneous
strain, polarization, or magnetization respectively are shown by red, yellow and blue
arrows in the figure. In multiferroics, additional interactions also arise (black arrows). For
example, in multiferroics which are simultaneously ferromagnetic and ferroelectric, a
coupling between these two orders (magnetoelectric coupling) may arise. In these
materials, called “magnetoelectric multiferroics”, electric polarization can be induced by
Chapter 1 INTRODUCTION & BACKGROUND
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 34
an external magnetic field, and vice versa (meaning magnetization can be induced by an
external electric field). It should be noted that not all ferromagnetic/ferroelectric
multiferroics exhibit magnetoelectric coupling [110].
The primary magnetoelectric (ME) effect is defined by the appearance of an electric
polarization under a magnetic field P(H): magnetoelectric (ME) output or a magnetization
at the application of an electric field M(E): electromagnetic (EM) output [110, 111]. The
former is also called the direct magnetoelectric effect (DME), describing magnetic field
induced polarization as shown in Eq. 1.25.
HP
α
= 1.25
while the latter is known as converse magnetoelectric effect (CME), describing electric
field induced magnetization as shown in equation Eq. 1.26 [112].
EM
α
= 1.26
This effect can be enabled due to direct coupling of electric field to magnetization,
magnetic field to polarization, polarization to magnetization, or indirectly via strain as
illustrated in Figure 1.15. The ME of second order (or secondary ME) consists on
variation of the permittivity under magnetic field ε(H) or the magnetic permeability at the
application of an electric field µ(E). Materials exhibiting ME properties can be single-
phase and composites (di- or poly-phasic systems). Single-phase multiferroics
intrinsically exhibit more than one ferroic order parameter. Typically either one or both of
the order parameters are weak and only arise at low temperatures. Multiferroic
heterostructures are artificially created by coupling two ferroic materials through an
interface.
The magnetoelectric effect was first postulated by Pierre Curie [113]. In 1959,
Dzyaloshinskii predicted this effect in Cr2O3 based on symmetry considerations and
Asrov confirmed this prediction experimentally in 1960 [111, 114, 115]. Many
investigations of this phenomenon were carried out in the 1960s and 1970s,
predominantly by Smolenskii [116] and Venevtsev [117]. The revival of interest in
magnetoelectric materials was initiated by N. Hill theoretical investigation of in 2000
[118] and by the recent discoveries of new mechanisms of ferroelectricity in perovskite
TbMnO3, hexagonal YMnO3, RMn2O5 (R= Tb, ), Ni2V3O8 and BiFeO3 [119-123].
Chapter 2 MOTIVATIONS & AIMS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 35
Chapter 2 MOTIVATIONS & AIMS
The intent of this chapter is to describe the motivation and aim of the present work.
Relaxor ferroelectrics have been introduced in the preceding chapter as ferroelectric
like systems that display a diffused phase transition as well as strong frequency
dependence in the magnitude of the dielectric constant and the peak temperature. It has
been observed that in the perovskites the relaxor behavior originates when the B-site atom
is replaced by other isovalent or heterovalent atoms [124]. The existence of random
dipolar fields has also been discussed as constituting an essential requirement for the
existence of relaxor behavior.
In the case of heterovalent substitution the random fields originate in the inherent
charge disorder due to the random distribution of B1 and B2 atoms on the B-sites of the
perovskite ABO3. This creates largely uncorrelated and quenched electric fields at the
sites of the ferroelectric-active ions as e.g., in the case of PMN. In the case of isovalent
substitution the relaxor behavior is understood to originate in the random bond breaking
in the system which itself is a consequence of the size difference between the B-atoms
[125]. If there is a sufficient number of broken bonds a redistribution of charges due to
local modification of the polarizability will give rise to (weak) quenched random dipolar
fields e.g. in Ba(Sn,Ti)O3 and Ba(Zr,Ti)O3.
Both of these (random field and random bond breaking) can give rise to the
formation of polar nano regions. It has been argued that relaxor behavior arises due to the
interaction between these polar nanoregions embedded in the highly polarizable matrix.
These interactions become stronger as temperature decreases. This behavior mimics the
well understood magnetic spin glass systems [1, 126]. Similar to the spin glass systems
where the competing effects between the spins are responsible for the glass behavior, in
the case of relaxors competing interactions develop between the PNRs leading to
frustration and hence to the observed frequency dependence in the dielectric response. In
the spin glass system this interaction is described in terms of order parameter [127-129].
In the formalism depicted by Edward-Anderson, the order parameter EA
q usually caters
the magnetic spin glass systems. Viehland et al, on similar grounds, have calculated the
order parameter for PMN a well-known relaxor ferroelectric, using SK model [130]. The
Chapter 2 MOTIVATIONS & AIMS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 36
temperature dependence of EA
q in the canonical/classical relaxor ferroelectrics is found to
be linear, as calculated by the Blinc and Pirc and experimentally verified by R. Blinc et al
[65, 131, 132]. However, the detailed behaviors of the relaxor state still a matter of
debate. In particular we wish to see how the said order parameter EA
q varies with
temperature in the isovalently substituted relaxor ferroelectrics. Equally importantly we
wish to explore the question of the temperature dependence of the dielectric response in a
dynamical situation εʹ(T, f) and whether this can also be represented meaningfully in
terms of a mean field order parameter such as EA
qwhich was originally defined as a time
independent entity.
With these general questions in mind we aim to study the development of the
relaxor properties in the zirconium substituted barium titanate system. We investigate
how the increasing concentration of Zr ions in the parent compound BTO modifies the
dielectric properties. We shall explore the question whether the changes introduced with
increasing Zr content can still be described in terms of a universal scaling behavior of the
form EA
q(T/Tm). We shall attempt to understand the higher temperature deviations from
the Curie-Weiss law in terms of correlations among the polar nano regions.
In a second part of our work we will investigate the behavior of a ferromagnetic
relaxor system in the diphasic composite material [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y. There is
considerable literature on ferromagnetic-ferroelectric materials produced by introducing
magnetic ions into ferroelectrics e.g., (1−x)Pb(Fe2/3W1/3)O3−xPb(Mg1/2W1/2)O3,
Pb2(CoW)O6 and B-site disordered Pb2(FeTa)O6 [133] which is ferroelectric and
antiferromagnetic, with weak ferromagnetism below 10 K. As a result of dilution of the
magnetic ions, these materials all have rather low Curie or Néel temperatures.
However a number of simple perovskite materials are known to have ferroelectric
and magnetic (mostly of the antiferromagnetic type) ordering. The interaction of the
ordered subsystems can result in ME effect, where the dielectric properties may be altered
by the onset of the magnetic transition or by the application of a magnetic field, and vice
versa. Due to their complex chemical structure, many of these systems are ferroelectric
relaxors, which are very interesting because of high dielectric, piezoelectric and
pyroelectric constants. A single phase multiferroics relaxor (CdCr2S4) with simple spinel
structure has been reported by J. Hemberger at. el. [134]. Composite materials containing
lead based relaxors system with ferromagnetic materials have also been reported. For
Chapter 2 MOTIVATIONS & AIMS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 37
example the PbTiO3-CoFe2O4 and Pb(Zr,Ti)O3-CoFe2O4 composite systems have been
studied by different groups. Levstik et al have reported the ferroic perovskite solid
solution 0.8Pb(Fe1/2Nb1/2)O3-0.2Pb(Mg1/2W1/2)O3 that exhibits both electric and magnetic
relaxor behavior. This is a site and charge order solid solution which shows broad and
frequency dependent maxima both in the electric as well as in the magnetic
susceptibilities [135]. Their results demonstrate that both the electric and magnetic
nanodomains exist in this system and that the observed magnetoelectric effect is due to
coupling between the local nano-cluster polarizations and magnetizations. C. Verma
reported the magnetoelectric coupling and relaxor properties of Pb0.7Sr0.3(Fe0.012Ti0.988)O3
thin films [136]. Arif D. Sheikh et al studied the magnetoelectric effect in
(f)Co0.8Ni0.2Fe2O4+(1-f)PMN-PT particulate composites [137]. There are few reports on
the barium based ferroelectric relaxor and ferromagnetic systems. M. Soda et. al. report
the superparamagnetism induced by polar nanoregions in (1-x)BiFeO3-xBaTiO3 relaxor
ferroelectric [138]. They examine the relationship between the relaxor like dielectric
property and magnetism in a single crystalline sample. From neutron diffraction results,
they found that the PNRs affect the magnetic ordering significantly, and the resulting
nano domains are the new origin of the superparamagnetism. P. Guzdek et al studied the
magnetic and magnetoelectric properties in the nickel ferrite-lead iron niobate relaxor
composites. They study the bulk ceramics and multilayer laminated structure of this
system [139].
In this thesis one of our goals is to explore the effect of a magnetostrictive
ferrimagnetic (FM) component (CoFe2O4) on the relaxor properties of Ba(Zr,Ti)O3 in the
composite material [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y. We shall also explore how applied
magnetic fields might have a significant effect on the relaxor properties of this system.
We expect the magnetic fields to affect the relaxor indirectly viz. the magnetostrictive
ferrite component can generate stresses that act on the relaxor component affecting its
behavior. A further interesting aspect is introduced by the randomness of the couplings
between the polar-nanoregions and the magnetic component. The randomly distributed
stresses originating in the ferromagnetic part may interact with the relaxor component
pushing it deeper into the relaxor regime. Furthermore, we explore if the effects of the
magnetic field on the FM-Relaxor composite can be described within the framework of
the mean field theories of spin glasses applied to relaxor systems.
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 38
Chapter 3 EXPERIMENTAL METHODOLOGY
It is intent of this chapter to describe the experimental techniques used to characterize
the bulk ceramics samples. Structural, electrical and magnetic properties of the
polycrystalline bulk can be determined using various methods. A detailed description
about synthesis and characterization tools used during the course of present work is
also given in the chapter.
3.1 Experimental Procedures
The overall experimental procedures for bulk ceramics are summarized in Figure
3.1.
Figure 3.1: The overall experimental procedure for the bulk ceramic.
Ferroelectrics and
Multiferroics Bulk
ceramics Materials
Structural Analysis
Electrical Properties
εʹ(T,f), ε″(T,f)
Magnetic Properties
M (H)
Processing of bulk ceramic materials
via conventional sintering processing
X-ray Diffraction.
Analysis of data from the measurements
using different theoretical models
Wayne Kerr 4275 LCR Meter
attached to He closed cycle
Cryostat Model CCS-350 and
homemade hi
g
h tem
p
erature setu
p
.
Vibrating Sample Magnetometer
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 39
3.2 Fabrication of Ferroelectric and Multiferroics Materials
The simplest and most common way of preparing solids is the ceramic method
(also known as solid state reaction method), which consists of heating together two
non-volatile solids which react to form the required product. The ferroelectric and
ferromagnetic ceramics material is prepared by the solid state reaction method. The
detailed steps of the fabrication process of ferroelectric, ferromagnetic and multiferroic
materials through solid-state reaction are summarized in Figure 3.2.
Figure 3.2: Various steps in conventional sintering method for processing of bulk materials.
Precursors
Chemicals
Mixing
Grounding thoroughly
and pelletization
Grinding
Calcination
Sintering
Crystalline Sample
Stoichiometric quantities of precursors
depend on the material formula.
Examples:
High purity chemical powders
Co2O3 and Fe2O3 for CoFe2O4
BaCO3, ZrO2 and TiO2 for
B
a
(
Zr,Ti
)
O3
May repeat these steps several times to
ensure a good crystalline sample
1. Grinding the mixed powder.
2. Calcined the
p
owder in the furnace.
Press the powder into the disc shaped
pellet that is suitable for experiments.
Sinter the pellet in the furnace.
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 40
The procedure is to take stoichiometric amounts of the binary oxides, grind them in
a pestle and mortar to give a uniform small particle size, and then heat in a furnace for
several hours in an alumina or porcelain crucible. An agate mortar and pestle was used for
the mixing and grinding of the precursors. Crucibles of porcelain and alumina
respectively were used for the heat treatment and synthesis of the powders.
3.2.1 Furnaces
We used two types of furnaces for heat treatment of the samples.
a) Box Furnace
The Chamber furnace model PLF 160/3 (Protherm) Alser Teknik, was used for the
calcination of the ceramic precursors.
b) Tube Furnace
The required sintering temperature of the studied sample was more than 1300 ˚C as
will be explained in chapter 4. The furnace used for this purpose is a tube furnace model
STF 16/180 (Carbolite) with a recrystallized alumina work tube. The maximum operating
temperature is 1600 ˚C for this particular tube furnace.
The tube furnace must be operated with a work tube, for example a recrystallized
alumina work tube 50 mm inner diameter and 900 mm long as supplied by Carbolite.
Stainless steel end cap with silicon rubber gasket can be inserted at both ends to close the
tube. The temperature is controlled by the temperature programmers Eurotherm model
2408CP. The Eurotherm model 2408CP is a digital instrument with PID control
algorithms which may be used as a simple controller or a 16-segment programmer.
3.3 X-ray Diffractometer
The crystal structure of bulk specimens was determined by studying the X-Ray
diffraction pattern. The diffractometer used was a PANalytical Empyrean system with
PreFix optical modules and fixed sample stage. The operating voltage was 45 kV and the
current was 40 mA. The source of X-ray is copper (Cu) with characteristic X-rays with
wavelengths Kα1~ 1.540593 Å, Kα2 ~ 1.54442 Å and Kβ ~ 1.54187 Å. Normal powder
diffraction, phase identification, quantitative analysis, high resolution direction are the
most common uses of these X-rays. Nickel (Ni) having K-absorption edge 1.4881 Å is
used as a beta filter in our instrument.
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 41
3.3.1 Powder Samples Configuration
Flat polycrystalline materials are analyzed using Bragg-Brentano geometry shown
in the Figure 3.3. In Bragg-Brentano para-focusing geometry, the incident X-ray beam
from the line focus of the X-ray tube diverges in the diffraction plane until it irradiates the
sample. The diffracted X-ray beam converges form the sample until it passes through the
receiving slit (the natural focusing point on the goniometer circle) before diverging again.
The optics used in the powder sample analysis are listed in Table 3.1. Two sets of
fixed divergence slits are supplied with the instrument. The slits marked 4º, 2º, 1º, 1/2º
and 1/4º are for the powder sample phase analysis. The other set is for the low angle
measurements, are used in thin film phase analysis or reflectometry applications. It
consists of three slits marked 1/8º, 1/16º and 1/32º. These slits must be inserted into the
slot for fixed divergence slits on the Prefix module.
Table 3.1: Powder sample configuration X-ray beam optics.
Incident Beam Optics Diffracted Beam Optics
Powder Sample Thin Film
Beam Attenuator Anti-Scatter Slit Parallel Plate Collimator
Beta Filter Receiving Slit Collimator Slit
Divergence Slit Detector Detector
Soller Slit
Beam Mask
Figure 3.3: Powder sample configuration for XRD measurements.
Divergence slits are used in Bragg-Brentano geometry for phase analysis to control
the divergence of the incident beam. The divergence angle must be such that the X-ray
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 42
beam is completely accepted by the sample. On the diffracted beam side an anti-scatter
slit is used with the acceptance angle the same as the divergence angle. For normal
powder diffraction, a fixed divergence slit (1º) is used to control the divergence of the
incident beam. Receiving slits (1/16º) are placed in the focusing point of the diffracted X-
ray beam. The opening of the slit determines the resolution of the measurement. The
receiving slit is used in combination with an anti-scatter slit (2º).
3.3.2 Thin Film Configuration
Phase analysis of thin films enables identification of phase(s) with increased surface
sensitivity, or depth profiling of multi-layer samples. In thin-film analysis the incident
angle of the X-ray beam is fixed at a very small angle with respect to the sample surface
determining the penetration of the X-rays, depending on the sample material and the
wavelength used. During the measurement only the 2θº angle changes. The thin film
configuration for the thin film phase analysis is shown in Figure 3.4.
Figure 3.4: Configuration for the phase analysis of thin film.
A combination of a very small divergence slit (1/4º or small) together with a parallel
plate collimator (0.27º) is used in the quasi-parallel beam geometry. In thin film analysis
the angle of incidence of the X-ray beam is fixed at very low angle with respect to sample
surface determining the penetration of the X-ray beam, depending on the sample material
and the wavelength used. During the measurement only 2θ angle changes. In reflectivity
measurements an ω-2θ scan is made at very low angles with respect to the sample
surface.
Ch
a
S
tu
d
3.4
cal
c
dif
f
Ca
p
do
w
con
t
die
l
die
l
sho
w
a
pter 3
d
ies of Ferro
e
Dielect
r
LCR Me
t
c
ulated fro
m
f
erent temp
p
acitance a
n
w
n in a clo
t
rolled by
l
ectric mea
s
Fi
g
3.4.1
Janis clo
l
ectric con
s
w
n Figure
3
¾
C
¾
C
¾
G
¾
V
¾
R
¾
T
e
lectric and
M
r
ic Prop
e
t
er was us
e
m
capacita
n
eratures.
T
n
d Dissipati
o
sed cycle
h
using tem
p
s
urement is
g
ure 3.5: Sch
e
Cryostat
sed cycle
r
s
tan
t
meas
u
3
.6. The m
a
C
ompresso
r
C
old Head
G
as Lines
V
acuum Jac
k
R
adiation S
h
T
emperatur
e
M
ultiferroic B
e
e
rties Me
a
e
d to meas
u
n
ce measur
e
T
he LCR
m
o
n factor
w
h
elium cry
o
p
erature co
n
shown bel
o
e
matic block
r
efrigerator
u
rements.
T
a
in parts of
t
(Model 82
0
k
et
h
ield
e
controlle
r
e
havior in [B
a
a
sureme
n
u
re the diel
e
ments ma
d
m
eter used
w
ere record
e
o
stat and t
e
n
troller. T
h
o
w in Figur
e
diagram of
t
(CCR) sy
T
he schem
a
t
he Janis C
C
0
0)
(
331 Lake
S
E
XPE
R
a
(Zr,Ti)O
3
]
1-
y
n
t
ectric prop
e
d
e in the fr
e
was the
W
e
d simultan
e
e
mperature
h
e block d
i
e
3.5.
t
he dielectric
stem mod
e
a
tic diagra
m
C
R system
i
S
hore)
R
IMENTAL
M
y
:[CoFe
2
O
4
]
y
S
e
rties. Diel
e
e
quency ra
n
W
ayne-Kerr
e
ously. The
cooling a
n
i
agram for
measureme
n
e
l CCS-35
0
m
of the
c
i
ncludes
M
ETHOD
O
S
ystem
e
ctric cons
t
n
ge 0.2-50
0
model 42
7
sample wa
n
d heating
r
complete
s
n
t setup.
0
was used
c
omplete s
y
O
LOGY
| 43
t
ant was
0
kHz
at
75. The
s cooled
r
ate was
s
etup of
for the
y
stem is
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 44
CCR system operates in the temperature range 10 K to 325 K and requires no liquid
helium. Helium gas is compressed and expanded based on Gifford-McMahon (M-G)
thermodynamic cycle using compressor (Model 8200) in a closed loop. During the
expansion phase of each cycle, heat is removed from the cold finger on which sample is
mounted. A TG-120-CU GaAlAs diode temperature controller and a 25 ohm heater are
installed on the cold finger to measure and control the temperature precisely. Prior to cool
down, the shroud is evacuated up to 1.0×10-4 torr using diffusion pump. Better vacuum
level provides greater insulation, resulting in shorter cool down times and lower final
temperatures.
Figure 3.6: Complete view of setup used for dielectric constant measurements.
3.4.2 Sample Holder
Sample was mounted on the sample holder made of brass with dimension of
mmmmmm 43016 ×× and attached at top end of cold finger as shown in Figure 3.7. Mica
sheet is attached on the brass sample holder which provides the better thermal contact
between sample and sample holder while insulating it electrically. Typical size of the
sample used to measure dielectric constant is circular pellet of diameter 10-12 mm. Two
copper leads were attached on the sample using silver paint. An additional TG-120-CU
GaAlAs diode thermometer is attached on the sample holder close to the sample for
accurate measurement of the sample temperature.
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 45
Figure 3.7: System wiring inside closed cycle system model CCS-350.
3.4.3 Temperature Controller (331 LakeShore)
The temperature controller model 331 of Lake Shore was used to control and
measure the temperature of the closed cycle refrigerator and the sample. Model 331
temperature controller has two sensor inputs with maximum heater output of 50 W for
sensor-A and 1 W for sensor-B. A TG-120-CU GaAlAs diode thermometer and 25 ohm
control heater attached at cold finger of closed cycle system were connected with input of
sensor-A and were used to control and measure the temperature of cryostat while sensor
attached on sample holder was contacted at input of sensor-B and was used to measure
the temperature of sample.
3.4.4 Dielectric Constant
Dielectric constant is a quantitative measure of the degree to which a medium can
resist flow of charge in the material. It (ε) is defined as
E
D
=
ε
3.1
where D is the electric flux density and E is the intensity of electric field. For the case of a
parallel plate capacitor of area A and spacing between the plates is d, and with the space
between the two plates filled with a dielectric material of permittivity ε, the capacitance
would be
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 46
d
A
Cr
0
ε
ε
= and A
Cd
o
r
ε
ε
= 3.2
The product
ε
ε
ε
=
0r is the called permittivity of the dielectric.
3.4.5 Dielectric Loss
Consider a parallel plate capacitor filled with a materials characterized by ε. With
the alternating fields the electric flux density can be expressed as
[]
)exp(Re)( tiEtD oro
ω
ε
ε
= 3.3
There is a variation of electric flux density with time, so there will be a flow of
current and the current density will be
[]
)exp(Re
)(
)( tiEi
dt
Dd
tJ oro
ωωεε
⋅== 3.4
Since εr is a complex quantity, we can write,
[]
)sin)(cos(Re)( titiiEtJ oo
ω
ω
ε
ε
ε
ω
+
′′
−
′
=
)sincos()( ttEtJ oo
ω
ε
ω
ε
ε
ω
′
−
′′
= 3.5
To derive a circuit analogue of the phenomenon of dielectric losses a parallel plate
capacitor with unit area of cross-section of the plates and separated by unit length. If the
applied voltage is, Eₒcosωt then from Eq. 3.5, which shows that, the dielectric may be
considered to be made up of a parallel combination of resistance and capacitance. If R is
the resistance and C is the capacitance of the dielectric,
tCE
R
tE
tJ o
o
ωω
ω
sin
cos
)( −=
with
ε
ε
ω
′′
=o
R/1 and
ε
ε
′
=o
Cwith the general expression of capacitance as (A = 1 m2
and d = 1 m)
d
A
Co
ε
ε
′
= 3.6
As εʺ→0 and R→∞.The loss tan
δ
is defined as
ε
ε
′′′ and is the measure of the dielectric
losses is
RC
ω
ε
ε
δ
1
tan =
′
′′
= 3.7
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 47
Where f
π
ω
2=is the applied frequency (in Hertz (Hz)), R and C are the equivalent
resistance (in Ohms ( Ω)) and capacitance (in Farads (F)) respectively.
3.4.6 High Temperature Dielectric Measurement System
For the high temperature measurement of dielectric constant, we designed and
construct the furnace and sample probe which is attached to the LCR meter for the
measurements.
a) Description of the Set-up
Functionally, the set-up consists of following parts: the sample holder, the main
heater and the temperature control unit, the pyrex vacuum tube and data processing
equipment, and the vacuum handling unit. Figure 3.8 shows final apparatus after
completion.
Figure 3.8: Photograph of high temperature dielectric measurement setup.
It consists of a pyrex glass tube having length 1.07 meter, outer diameter 24.1 mm
and inner diameter 19 mm. One end of the tube is closed, and the other end is attached to
the specially designed aluminum coupling using epoxy, this Al assembly consists of
vacuum port for the mechanical pump, and vacuum port for the probe head of the sample
holder. The cross-sectional views with dimension of this part and probe head of the
sample holder are shown in Figure 3.9. The pyrex glass tube clamped on the iron stand.
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 48
Figure 3.9 : The cross sectional view of the vacuum ports for vacuum pump and sample holder probe head. All the above assembly is made
of aluminum. The vacuum port for the pump is fixed in the port of 10 mm with an O-ring. . All the dimensions are in millimeter.
Port for glass
tube coupling
Prob head attachment
port
Port for pin connector
Port for vaccum
coupling
Prob head of Sample Holder Glass Tube End Coupling
Vacuum Port for Mechanical Pump
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 49
a) Heating Element
Heater was wound over pyrex glass tube, that can sustain temperature up to 600 ºC.
The heating element is capable to produce temperature up to 500°C. The power of the
heating element is 1000 W and is operated on 220 Volts. The wire used for the elements
is 23 standard wire gauge (SWG) Nichrome wire of diameter 0.61 mm
)(/
2WRVP = and )(
4
2Ω= d
l
R
π
ρ
3.8
Resistivity of the nichrome wire is 1.016×10-6 Ω-m, and the required length of the
wire for the element was calculated using Eq. 3.8. The required length of the nichrome
wire for the element is 14 meter, and is wound in the form of coil with inner diameter 2.5
mm and length of the element in the form of the coil is ~2.2 m. A pyrex glass tube having
diameter 24.1 mm used to wound the element on it, and the total number of turn in the
element are 22 of the nichrome wire coil. The length of the heating element wound is 230
mm (approximately 9 inch). Ceramics beads are used to avoid the short circuiting of the
heater coil. Figure 3.10 shows the schematic diagram of heating element winding.
Figure 3.10: Schematic diagram of the heating element.
b) Sample holder
The sample holder is made of a pyrex glass tube having diameter 6 mm and length
550 mm. On one end of the tube rectangular flat pyrex glass piece (20 mm × 12 mm × 2.5
mm) is attached by means of glass blowing which act as a sample holder and the other
end is fitted to the probe head with teflon seal and screwed in the probe head. The sample
holder is shown in the Figure 3.11. K-type thermocouple is attached to the sample holder
to read the temperature. The wires for the measurements are passed through the tube.
230 mm
279.4
mm
1070 mm
Pyrex Glass Tube
Insulating Brick
Ceramic Beads
Coil element
Ch
a
S
tu
d
Fig
u
(fo
r
the
thr
e
co
m
soc
k
the
pro
v
inc
r
inc
r
a
pter 3
d
ies of Ferro
e
Figure
3
c)
The com
m
u
re 3.12. T
h
r
dimensio
n
pin conne
c
e
aded four
h
m
ing from
t
k
et of the c
o
d)
The tem
p
furnace. I
T
v
ide up to
r
eased by
r
ease the
c
e
lectric and
M
3
.11: Sample
Vacuum
T
m
ercial eig
h
h
e port for
n
s see Figur
e
c
tor port u
s
h
oles paral
l
t
he sample
o
nnector.
Figure
Temperat
u
p
erature co
n
T
C502 can
80 W of
using the
c
urrent in
M
ultiferroic B
e
Holder desi
g
T
ight Conn
e
h
t pin con
n
the vacuu
m
e
3.9). The
s
ing a gas
l
el to the h
o
and therm
o
3.12: The co
m
u
re Contr
o
n
troller mo
accept a
w
heating p
o
magnetic
c
the heate
r
e
havior in [B
a
g
ned for high
e
ctor
n
ector used
f
m
tight con
n
fixed pane
l
kit for va
c
o
les in the
o
couple ar
e
m
mercial va
c
o
ller
(ITC
5
del ITC50
2
w
ide range
o
o
wer. Heat
i
c
ontractor
r
.
K-type
E
XPE
R
a
(Zr,Ti)O
3
]
1-
y
temperatur
e
f
or the elec
t
n
ector is al
s
l
socket is
f
c
uum arran
g
fixed pane
l
e
soldered
o
c
uum tight p
i
5
02 Oxfo
r
2
was used
o
f differen
t
i
ng power
with the t
thermocou
p
R
IMENTAL
M
y
:[CoFe
2
O
4
]
y
S
e
dielectric m
t
ric connec
t
s
o made of
f
ixed on th
e
g
ement. T
h
l
socket. A
l
o
n the inn
e
i
n connector
r
d)
to control
t
temperat
u
of the te
m
emperature
p
le is use
d
M
ETHOD
O
S
ystem
easurements
t
ion is sho
w
aluminum
e
circular s
u
h
e circular
p
l
l the elect
r
e
r part of t
h
the tempe
r
u
re sensors
m
perature c
o
controlle
r
d
to mea
s
O
LOGY
| 50
.
w
n in the
cylinder
u
rface of
p
ort has
r
ic wires
h
e panel
r
ature of
and can
o
ntroller
r
, which
s
ure the
Chapter 3 EXPERIMENTAL METHODOLOGY
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 51
temperature. Connections of heater and sensor are made by means of a 9 way D-socket on
the rear panel of ITC502. The normal way in which ITC502 affects its control, is by
applying power to a heater. There are two ways to control the heater voltage i.e. manual
or automatic. In manual control the heater voltage may be varied using the controls on the
front panel of ITC502 while in automatic control, the heater voltage is varied in response
to the difference between a measured temperature and a set point. The criteria for good
control of temperature;
1) The mean temperature of the system should be as close as possible to the desired
temperature.
2) The fluctuations above and below the mean temperature should be very small.
3) The system should follow changes in the set point as rapidly as possible.
For accurate temperature control avoiding too slow a response or overshoot we can
use proportional, integral and derivative control settings of the temperature controller.
3.5 Vibrating Sample Magnetometer
Magnetization measurements were made using a commercial vibrating sample
magnetometer (VSM) model BHV-50 of Riken Denshi Co. Ltd. Japan. The sample is
attached to the vibrator by means of a teflon coupling and a straw. The vibration is along
the z-axis with fixed amplitude of 5 mm and a frequency of 30 Hz. The magnetic field is
applied along the x-axis by means of an electromagnet capable of producing fields up to
~7 kOe. The magnetic moment of the vibrating sample induces an electromotive force in
the pick-up coils arranged along the x-axis. The sample response is calibrated using
standard nickel samples (Ni 2.441 emu). The sensitivity of the system is 10-4 emu.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 52
Chapter 4 PROCESSING AND STRUCTURAL
CHARACTERIZATION
In this chapter we describe the synthesis and structural characterization of the
prepared bulk ceramics.
4.1 Synthesis Techniques
Ceramic powder synthesis is an important technology in chemical and ceramics
engineering. Any synthesis technique should give uniformity in the microstructure of a
single phase ceramic for enhanced properties. Numerous techniques are available in the
literature for the synthesis of ceramics. Selection of the synthesis route is crucial to
control the composition, structure, and morphology of a chosen material. There are
mainly two approaches for synthesis of ceramic powder; the chemical method and the
mechanical method. Mechanical methods are (a) mixed oxide process or solid state
reaction process and (b) high energy ball milling (ball mill, planetary ball mill, rotator
ball mill, etc.). The chemical methods of synthesis of ceramic powders are sol-gel
methods, co-precipitation method, hydrothermal method, combustion method, molten
salts, liquid-phase and gas-phase reactions, polymer pyrolysis, pechini method, citrate gel
methods, aerosols and emulsions etc. [140]. The chemical method gives a better product
as compared to the mechanical method by the absence of secondary phase and chemical
and structural homogeneity, but the main disadvantage of chemical method is that it is
time consuming, reaction procedures are complex and generally requires of costly
ingredients. The initial product obtained from chemical method is also calcined at
temperatures 500 ˚C to 1000 ˚C depending on the material which is same as in solid state
reaction route. The chemical precursors taken in chemical method are hydrophobic,
unstable at room temperature, and hence react with other materials.
The disadvantage of solid state reaction assisted synthesized powders has imperfect
surface structure, non-uniform strain and coarse particle size due to prolonged heating at
high temperature [141]. Still the solid state reaction route method is well suited for large-
scale production of bulk ceramic powders. It requires low cost precursors which are
readily available and the preparation techniques are much easier.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 53
4.2 Solid State Reaction Method
The solid state reaction route is the most convenient and widely used method for the
preparation of polycrystalline solids from a mixture of solid oxide and carbonates as
precursors. Heat is required for solid precursors to proceed for reaction, as generally
solids do not participate in reaction at room temperature. The thermodynamic free energy
determines the feasibility of the reaction of precursors. The rate of reaction depends upon
reaction conditions, structural properties of the reactants, surface area of the solids and
their reactivity [142]. In solid state synthesis the reactants are heated for a variety of
reasons. The precursors have to be thermally decomposed to produce the required
fragments for direct combination. In addition, the higher temperature allows some
movement or flow of atoms through the solid at a sufficient rate so that the desired
product can eventually be obtained. This process is enhanced if one of the components
melts, thus overcomes the ‘solid state diffusion barrier [143]. This barrier arises because
the reactions can only occur between neighboring atoms. In a solid, the atoms have to
migrate through the rigid solid lattice. This process is slow unless the temperature is
raised significantly to allow rapid migration. A number of procedures are used to reduce
the time needed for synthesis. The starting materials are often intimately grinded together,
thus ensuring good mixing and hence increasing contact between the reacting grains.
4.2.1 Precursors
Precursors are the raw materials for the reaction from which the required solid
crystalline compound will form as product. The nature of raw material has a major effect
on the properties of the final ceramic material. The quality of raw materials depends upon
the purity percentage and particle size. The reagents are selected on the basis of reaction
conditions and the nature of the product. The reactants are dried thoroughly before
weighing, to remove the moisture. The surface area of reagents influences the reaction
rate for which fine grained materials should be used. If the raw materials contain some
impurities, it will affect the physical properties of the final product material [144].
The precursors used for the preparation of ferroelectric, relaxor ferroelectrics and
ceramic ferrites are given in the Table 4.1. All these oxide/carbonates powders were
having more than 99% purity. Polyvinyl Alcohol (C2H3*) is used as binder for the
pelletization of the samples.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 54
Table 4.1: The precursors used for the bulk powder ceramic processing.
Chemicals Formula Purity
(%)
Molecular
Weight Supplier
Barium Carbonate BaCO3 >99 197.35 Avonchem
Titanium (IV) Oxide TiO2 99-100.5 79.87 Riedel-de Haёn
Zirconium (IV) Oxide ZrO2 >99 123.22 Fluka
Cobalt (II-III) Oxide Co3O4 99.9 240.80 Fluka
Iron (III) Oxide Fe2O3 99.99 159.69 Aldrich
Nickel Oxide NiO 76%Ni 74.69 Fluka
Polyvinyl Alcohol C2H3* Riedel-de Haёn
4.2.2 Weighing, Mixing and Grinding
The reactants were taken in the stoichiometric ratio for a desired compound to form.
There are two ways to calculate the stoichiometry of the precursor. First write the balance
chemical equation and from the left hand side of the chemical equation calculate the
required masses of the precursors.
231223 )1( COOBBAOBxOBxACO xx +
′′′
→
′′
+
′
−+ − 4.1
Let ‘M’ be the molecular weight of the desired ceramic and ‘m’ be the amount of
prepared material. ‘Ma’ is the molecular weight of the ath metallic oxide/carbonate used in
the synthesis of the ceramic and ‘x’ is the fraction of “a” metallic ion in the ceramic. Then
weight ‘ma’ required for ath metallic oxide/carbonate is given by
M
mxM
ma
a= 4.2
These precursors are manually mixed in an agate mortar and pestle. Some organic
volatile liquid (preferably acetone or alcohol) is added to the mixture and grinded till it
dried. The liquid is used for a homogenous mixture of precursors to avoid formation of
secondary phase. The preferred amounts of precursors were thoroughly mixed in an agate
mortar, with suitable amount of acetone (volatile organic liquid) for 2 to 3 hr. The acetone
gradually vaporized during the process of mixing and grinding.
4.2.3 Calcination
The heating of the mixture depends on the form and reactivity of the reactants.
Calcination is used to achieve the desired crystal phase and particle size. For the heating
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 55
of material a chemically inert and high melting point container is used, generally porcline
and alumina. The calcination process is an endothermic decomposition reaction which
gives oxide as a solid product and liberates gases [145].
4.2.4 Green Body Formation
After calcination the fine powder was reground. After drying, the powder is
compacted by dry-pressing in steel dies to obtain the green body in the desired, final
form, but with slightly greater dimensions, because during the next operation, sintering,
shrinkage takes place. To facilitate compaction and to increase the strength of the green
body, a binder is usually added before pressing.
Polyvinyl alcohol (PVA) used as a binder, was added at 5 at% to improve the green
strength of the powder compacts. After adding the binder to the reground calcined
powder, we were pressed into pellets under a uniaxial pressure of 4×106 kg/m2 for 15
minutes.
4.2.5 Sintering
Sintering is the removal of pores between the calcined particles by the shrinkage of
the powder as well as the growth of particle and formation of the strong bond between
adjacent particles. The powder is mixed with a binder and then compacted into a pellet
form using a hydraulic press and is used for sintering. The binder will burn out at the time
of sintering. The sintering transforms the pellet to a strong, dense ceramic body with
closely packed grains and randomly distributed crystallographic orientation. The
reduction of excess energy associated with the surface is the driving force of the sintering
process [146]. The solid/vapor interface becomes solid/solid interface which gives rise to
the grain boundary area and the grain growth mechanism gives densification to the
sample. The sintering temperature is always higher than calcination temperature.
4.2.6 Electroding
The sample surfaces were polished for smoothness and then silver paint or any
conducting paint is used as an electrode. The material is now placed between two
electrodes as a dielectric material between parallel plate capacitor. The electroding
material should adhere to the sample surface perfectly. The electrode should have zero
electrical resistance. The conducting paint should be in thin layer form. The electrodes
also can be made by a deposition method using sputtering. Also conventional
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 56
photolithography and chemical etching techniques were employed to make the desired
electrode.
4.3 Synthesis of Relaxor Ferroelectrics Ba(Zr,Ti)O3
Ferroelectric BaTiO3 and relaxor ferroelectric compositions BaZrxTi1-xO3 (0.3 ≤ x ≤
0.8) were prepared by the conventional solid state reaction [97, 147]. Reagent grade,
barium carbonate (BaCO3), titanium oxide (TiO2) and zirconium oxide (ZrO2) were taken
as the initial raw materials for the preparation of BaZrxTi1-xO3 relaxor ferroelectric
ceramics. The detail about the precursors is given in Table 4.1. Stoichiometric
composition of BaZrxTi1-xO3 ceramics with 0.3 ≤ x ≤ 0.8 were prepared by thoroughly
mixing the stoichiometrically weighed oxides and carbonates powders. The samples
prepared are as follow;
The general chemical reaction of BaZrxTi1-xO3 (0.3 ≤ x ≤ 0.8) in equation form is
given below.
231223 )1( COOTiBaZrxZrOTiOxBaCO xx +→+−+ − 4.3
The ratio of molecular weights of the precursors used is calculated using above
equation for different values of “x” and are listed in the Table 4.2. Based on the
geometrical packing of atoms, Goldschmidt developed the perovskite tolerance factor t
[148], for the solid solution with formula AB11-xB2xO3 is given by the equation below
()
[]
OBB
OA
SS rxrrx
rr
t++−
+
=
21
12 4.4
where rA, rB1, rB2 and rO are the respective ionic radii. The tolerance factor allows us to
estimate the degree of distortion. Generally it is an excellent starting point to determine is
a given combination of ions will form stable perovskite structures. It has been reported
that the stability of the perovskite structure may be expected within the limit of (0.88 < t
< 1.09), using radii corrected for cations coordination numbers. Table 4.2 also shows the
values of tolerance factor “t” calculated using the Eq. 4.4.
The preferred amounts of precursors were thoroughly mixed in an agate mortar with
suitable amount of acetone (volatile organic liquid) for 2 to 3 hr. The volatile organic liquid
was added in the mixture for homogenization of required phases. The acetone gradually
vaporized during the process of mixing and grinding.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 57
Table 4.2: The ratio of molecular weight of the precursors for Ba(Zr,Ti)O3.
-x Formula -t Stoichiometric ratio of precursors
BaCO3 TiO2 ZrO2
0.0 BaTiO3 1.058 7.89344 3.19512 0.0
0.3 BaZr0.3Ti0.7O3 1.040 7.89344 2.23660 1.47862
0.4 BaZr0.4Ti0.6O3 1.034 7.89344 1.91709 1.97150
0.5 BaZr0.5Ti0.5O3 1.029 7.89344 1.59757 2.46437
0.6 BaZr0.6Ti0.4O3 1.023 7.89344 1.27806 2.95725
0.7 BaZr0.7Ti0.3O3 1.017 7.89344 0.95855 3.45012
0.8 BaZr0.8Ti0.2O3 1.012 7.89344 0.63903 3.94300
After needed and apt mixing, mixed oxide powders of BaTiO3 and relaxor compositions
BaZrxTi1-xO3, (0.3 ≤ x ≤ 0.8) were calcined at temperature 1200 ˚C for 2 hours inside a box
furnace in a porcline crucible at heating rate 5 ˚C/min and then cooled (Figure 4.1).
Figure 4.1: Heat treatment cycle and procedure for magnetic ferroelectric ceramic Ba(Zr,Ti)O3.
The calcined powders were reground to fine powder by an agate mortar for breaking
the agglomerates. Calcination is to avoid the additional undesirable phases in the final
materials. The above calcined powder was reground and mixed with 5 at% polyvinyl
alcohol (PVA) as a binder in mortar and pestle. The binder mixed powder was compacted
to form pallet by a hydraulic press at 4×106 kg/m2 pressure using die set.
1200
1500
Time (hrs)
Tem
p
erature
(
˚C
)
2 hrs
5 hrs
5 ˚C/min
5 ˚C/min 5 ˚C/min 3 ˚C/min
Powder Form Pellet Form
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 58
The pellets of compositions BaZrxTi1-xO3 (0.3 ≤ x ≤ 0.8) were taken on an alumina
crucible and sintered at 1500 ˚C for 5 hours in a programmable tube furnace at heating rate of
5 °C/min (Figure 4.1). The added binder was burnout at 300–400 ˚C during sintering. The
pellets of the BaTiO3 were taken on an alumina crucible and sintered at 1400 °C for 5 hours
to avoid the melting. The appropriate sintering of the pallets is vital for electrical
measurement. Few of the sintered pellets were reground and used for the study of their
phase formation by means of X-ray diffraction. Theses pellets can also be use as a target for
the pulsed laser deposition for thin film preparation.
The sintered pallets were polished by emery paper and painted with silver paste as
an electrode for electrical measurement. The painted samples were kept in the oven for 2
hours to evaporate the moisture in the sample.
4.4 Synthesis of Magnetic Ferrites CoFe2O4
Ferrites can be prepared by almost all the existing techniques of solid state
chemistry. Some of these methods have been developed to prepare ferrites with specific
microstructures. The oldest one, the ceramic method, involves the same operations as the
classical techniques for fabrication of conventional ceramics. We used the solid state
reaction method to prepare the ferrite ceramics sample [149].
For the cobalt ferrite, cobalt oxide (Co3O4) and iron oxide (Fe2O3) were used as
initial raw materials. Co3O4 was used to introduce Co. The samples prepared are as
follow,
The chemical reaction is given below.
2423243 5.033 OOCoFeOFeOCo +→+ 4.5
The ratio of molecular weight of the precursors used is
Co3O4 : Fe2O3
4.81594 : 9.58153
The powders in the desired ratio were mixed in a mortar and grounded thoroughly
with acetone for two hours; the well-mixed powder was calcined at 900 ˚C in air for 5
hours. The calcined powders were then ground again for two hour and calcined again at
900 ˚C in air for 5-hours. The calcined powders were then ground again for two hour and
then the powder was pressed into disc shaped pellets by using a uniaxial hydraulic press
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 59
under 4×106 kg/m2 of pressure. The pellets of compositions CoFe2O4 were taken on an
alumina plate and sintered at 1300 ˚C for 5 hours in a programmable tube furnace at
heating rate of 5 ˚C/min in air and allowed to furnace cool at the rate of 3 ˚C/min. Figure
4.2 show the heat treatment cycle of CoFe2O4 ferrites. These pellets were also used as a
target for the pulsed laser deposition.
Figure 4.2 Heat treatment cycle and procedure for magnetic ceramic CoFe2O4 sample.
4.5 Synthesis of Composite Ceramics [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y
In the multilayer co-fired structure consisting of ferroelectrics and ferrites layers,
there are always many undesirable defects, such as cracks, pores and cambers, owing to
the co-firing mismatch between different material layers, which will damage the property
and reliability of end products [150].
Many material systems, such as BaTiO3-Ni/Cu/Zn ferrite, BaTiO3-Mg/Cu/Zn
ferrite, Pb(Zr0.52Ti0.48)O3-Ni/Cu/Zn ferrite, Pb(Mg1/3Nb2/3)O3-Pb(Zn1/3Nb2/3)O3-PbTiO3-
Ni/Cu/Zn ferrite and Bi2(Zn1/3Nb2/3)2O7-Ni/Cu/Zn ferrite, were investigated and found to
exhibit fine dielectric and magnetic properties. In these reports, spinel ferrites, such as
Ni/Cu/Zn ferrite, were always used as the magnetic phase of composite ceramics, because
they are mature materials for co-fired inductive components.
We chose the pure cobalt ferrite for the magnetic component system and
Ba(Zr,Ti)O3 as relaxor ferroelectric part to study the effect of magnetic field on the
relaxor behavior of the Ba(Zr,Ti)O3 in the composite [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y. Due to
the different sintering temperatures and shrinkage rates of ferroelectric phase and
900
Tem
p
erature
(
˚C
)
Time (hrs)
5 ˚C/min 5 ˚C/min 5 ˚C/min
3 ˚C/min
Pellet Form Powder Form
1300
Powder Form
5hrs 5hrs
5hrs
5 ˚C/min 5 ˚C/min
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 60
ferromagnetic phase, remarkable co-firing mismatch often occurs and results in
undesirable defects, such as cracks and cambers. As a result, the property of composite
ceramics and the reliability of end products are damaged. Thanks to the existence of large
amount of grain boundaries to dissipate stress, the composite ceramics with powder
mixture have much better co-firing behavior than the multilayered composite ceramics.
Although the mismatch of densification rate is alleviated to a larger extent, a good
sintering compatibility between ferroelectric and ferromagnetic grains is still required for
better co-firing match. The starting temperature of shrinkage and the point of maximum
shrinkage rate are both important for the co-firing behavior of composite ceramics. Some
research indicates that the composite ceramics exhibits an average sintering behavior
between two phases and the shrinkage rate curve of composite ceramics is between those
of two component phases [151].
Figure 4.3: Heat treatment cycle and procedure for composite [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y samples.
The composites with compositional formula [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y with y =
0.35 were prepared by mixing the two phases. The mixing process was carried out by
agate mortar and pestle in acetone. After drying, a small amount of 5 at% polyvinyl
alcohol (2-3 drops) was added to the powder mixture as a binder which was pressed into
circular discs under 4×106 kg/m2 of pressure using a uniaxial hydraulic press. The pellets
were finally co-fired at 1400 ˚C for 5 hours with a heating rate of 5 ˚C/min in air and
allowed to furnace cool at the rate of 3 ˚C/min. Figure 4.3 shows the heat treatment cycle
of [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y for y = 0.35 composite sample.
1400
Time (hrs)
Tem
p
erature
(
˚C
)
5 hrs
5 ˚C/min
3 ˚C/min
Pellet Form
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 61
4.6 Structural Analysis
The constructive and destructive interference of coherently scattering radiations is
controlled by geometry at the interatomic level and is governed by well-known Bragg’s
law
θ
λ
sin2dn = 4.6
where θ is the Bragg’s angle, λ is the wavelength of the X-rays and d is the interplanar
spacing. The interplanar spacing for the hkl plane of a cubic and tetragonal systems are
[152]:
()
Cubic
a
lkh
dhkl 2
222
2
1++
= 4.7
()
Tetragonal
c
l
a
kh
dhkl 2
2
2
22
2
1+
+
= 4.8
where h, k and l are the Miller indices of the planes of atoms. Bragg’s condition requires
that a suitable combination of λ and θ be found for efficient reflection.
Powder X-ray diffraction (XRD) measurements were performed at room
temperature over the range 2θ = 20−80° with a PANalytical Empyrean diffractometer
using Cu Kα radiation (λ = 1.5406 Å) and the Bragg−Brentano θ-θ configuration. The
data is taken as a continuous scan type having step size of 0.02˚ degree and step time of
three seconds.
4.6.1 Structural Characterization of Ba(Zr,Ti)O3
The purity and crystallinity of the synthesized samples were examined by the
powder XRD technique. The room temperature XRD for the BaTiO3 ceramic sample is
shown in Figure 4.4. The pattern indicates that the synthesized material shows good
agreement with the conventional tetragonal BaTiO3 structure (PCPDF data No. 891428),
with no impurity peak appearing in the diffractogram. The lattice constant was
determined by using Eq. 4.8 from the XRD pattern are a = 3.991 Å and c = 4.013 Å. The
volume of the lattice a2c is 64.2725 Å3.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 62
20 30 40 50 60 70 80
(103)
(101)
(001)
(113)(311)
(310)
(202)
(211)
(210)
(102)
40 42 44 46 48 50
(200)
(002)
(301)
(300)
(212)
(220)
(112)
(201)
(200)
(002)
(111)
(110)
(100)
Intensity (a.u)
Angle 2
θ
(degree)
Figure 4.4: The room temperature X-ray diffractogram of BaTiO3.
The XRD patterns were indexed as single phase BaTiO3 perovskite, with a clear
tetragonal symmetry characterized due to the visible splitting of the peak located at 2θ =
45° (Figure 4.4 inset). In general, peak at 45° is split into two peaks at 44.918° and
45.388°. These peak correspond to the (hkl) Miller index (002) and (200), whereas cubic
BaTiO3 (PCPDF data No. 892475) has one single peak at 45.096° corresponding to (200).
Therefore, we can conclude that the synthesized BaTiO3 powders show a tetragonal
structure.
The room temperature X-ray diffraction spectra of sintered Zr doped barium titanate
(BaZrxTi1-xO3) samples are shown in Figure 4.5. All the samples were found to be single
phase, with no additional phase and polycrystalline with no preferred orientation. The
peaks in the XRD patterns can all be indexed and correspond to a cubic perovskite
structure. The patterns are in good agreement with the standard card PCPDF card #
360019.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 63
20 30 40 50 60 70 80
(311)
(310)
(220)
(211)
(210)
(200)
(111)
(110)
(100)
BZ0.8
BZ0.7
BZ0.6
BZ0.5
BZ0.4
BZ0.3
Intensity (a.u)
Angle 2
θ
(degree)
Figure 4.5: The X-ray diffractogram of Ba(Zr,Ti)O3 (0.3≤ x ≤0.8).
It is observed form the XRD-pattern that the principal diffraction peak shift toward
the lower angle with increasing of Zr content. This shift is due to the incorporation of
larger ionic radii Zr4+ (0.72 Å) in place of smaller ionic radii Ti4+ (0.605 Å). The strains
developed due to the difference on size of Zr4+ and Ti4+ cations. This is clear indication
that the Zr is systematically dissolved in the BaTiO3 lattice in the studied compositions.
No splitting of (200) peak is observed which reveal that the structure is cubic perovskite.
The lattice constant of the sample was determined by the Eq. 4.7 and are listed in
Table 4.3. This data show an increase in the lattice constant with increase in Zr content in
the samples, the lattice is expand due to the larger size of Zr ion. Figure 4.6 shows the
graphical representation of the data in Table 4.3. The x axis is the molecular fraction of
the atoms substituted for titanium atoms in solid solution. The y axis is the lattice
parameter calculated from the X-ray diffractogram of the solid solution for different
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 64
concentration of zirconium. The linear increase in the lattice constant is consistent with
Vegard’s law [153].
Table 4.3: Values of the lattice parameter of Ba(Zr,Ti)O3 calculated from the XRD diffractogram.
-x Formula Abbreviation Lattice Parameter (Å)
0.3 BaZr0.3Ti0.7O3 BZT0.3 4.062 ± 0.001
0.4 BaZr0.4Ti0.6O3 BZT0.4 4.081 ± 0.001
0.5 BaZr0.5Ti0.5O3 BZT0.5 4.099 ± 0.002
0.6 BaZr0.6Ti0.4O3 BZT0.6 4.118 ± 0.002
0.7 BaZr0.7Ti0.3O3 BZT0.7 4.138 ± 0.001
0.8 BaZr0.8Ti0.2O3 BZT0.8 4.154 ± 0.001
0.30.40.50.60.70.8
4.04
4.06
4.08
4.10
4.12
4.14
4.16
Lattice Constatnt a
Vegards's Law Fit to a- Data
Zr Concentration x
a (Å)
67
68
69
70
71
72
73
Lattice Volume V
Vegard's Law Fit to V- Data
V (Å3)
Figure 4.6: Variation of lattice constant “a” and lattice volume “V” with Zr concentration in the
Ba(Zr,Ti)O3 (0.3≤ x ≤0.8).
The simplest mathematical expression of Vegard’s law is
BZOBTOBZT xaaxa +−= )1(
where aBZT is the lattice parameters of the mixture and aBTO and aBZO are the lattice
parameters of its components and x is the mole fraction of the second component [154].
The more correct formula in three dimensions would be
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 65
333 )1( BZOBTOBZT xaaxa +−=
Strong linear correlations for the lines developed from this data are obtained, R2 =
0.99954 and 0.99966 for the “a” and “V” curves. For the prepared solid solution
containing BaTiO3 and BaZrO3 the equation for the line generated from the graph yields:
()
xTiZrya18676.000598.4, +=
()
xTiZryV45741.917172.64, +=
where ya(Zr,Ti) is the lattice parameter and yV(Zr,Ti) is the lattice volume for the solid
solution and x is mole fraction of Zr in the system. The values of the lattice constant for
the end products are found form the fitting parameter. The value of aBTO and aBZO are listed
below.
Table 4.4: Values of the lattice parameter of BaTiO3 and BaZrO3 calculated from data.
Formula Lattice Parameter a (Å) Lattice Volume V (Å3)
BaTiO3 4.006 ± 0.001 64.27 ± 0.05
BaZrO3 4.193 ± 0.001 73.62 ± 0.04
These values are in very good agreement with the reported values of the lattice
parameter of BaTiO3 and BaZrO3 [155, 156].
The X-ray density or theoretical density was estimated using the relation
∑
=VN
A
A
XRD .
ρ
4.9
where A is the atomic weight of all the atoms in the unit cell, V is the unit cell volume and
N is the Avogadro’s number [152]. Since each primitive unit cell of the perovskite
structure contains one molecule, the X-ray density was determine according to the
following relation
3
.aN
M
A
XRD =
ρ
4.10
where M is the molecular weight of the particular perovskite, a3 is the volume of the
cubic unit cell. The variation in ρXRD as function of Zr concentration is shown in Figure
4.7. From Figure 4.7 it is observed that the ρXRD increases with the addition of Zr4+ ion
content, which is also attributed to the ionic radii of constituent ions causing increase in
lattice parameter.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 66
0.30.40.50.60.70.8
6.08
6.10
6.12
6.14
6.16
6.18
6.20
6.22
ρ
XRD (g/cm3)
Zr Concentration x
Figure 4.7: Variation of XRD density ρXRD with Zr concentration in the Ba(Zr,Ti)O3 (0.3≤ x ≤0.8).
4.6.2 Structural Characterization of CoFe2O4
The X-ray diffractogram of the cobalt ferrites ceramics are shown in the Figure 4.8.
20 30 40 50 60 70 80
(622)
(620)
(511)
(440)
(533)
(422)
(400)
(222) (311)
(200)
(111)
Intensity (a.u)
Angle 2
θ
(degree)
Figure 4.8: X-ray diffractogram of CoFe2O4.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 67
The peaks correspond to the expected inverse spinel structure. These peaks are
indexed to the cubic CoFe2O4 phases according to the standard PCPDF Card # 791744.
No un-indexed peaks were observed in the prepared samples.
The lattice parameter values are calculated form the X-ray diffraction pattern are
8.385±0.003 Å for CoFe2O4. These values are in good agreement with the reported
values. The X-ray density of the ferrite samples were calculated using the Eq. 4.9. Since
each primitive unit cell of the spinel structure contains eight molecules, the X-ray density
was determined according to the following relation [157].
3
.
8
aN
M
A
XRD =
ρ
4.11
The values of ρXRD are 5.287±0.006 g/cm3.
4.6.3 Structural Characterization of [Ba(Zr,Ti)O3]0.65:[CoFe2O4]0.35
Figure 4.9 depicts the X-ray diffraction patterns of [Ba(Zr,Ti)O3]0.65:[CoFe2O4]0.35
composite ceramics.
20 30 40 50 60 70 80
(310)
(220)
*(511)
*(122) *(122)
*(422)
(210)
(200)
*(400)
(111)
*(222)
(311)
(110)
*(200)
(100)
(113)
[BZ70]65:[CFO]35
[BZ60]65:[CFO]35
[BZ50]65:[CFO]35
Intensity (a.u)
Angle 2
θ
(degree)
[BZ40]65:[CFO]35
Figure 4.9: X-ray diffractogram of [BZTx]0.65:[CFO]0.35 (0.4≤ x ≤0.7).
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 68
From the figure it is clear that the composites have both Ba(Zr,Ti)O3 and CoFe2O4
phases. The peaks corresponding to BZTx and CFO were identified from their individual
patterns. All the peaks could be identified; hence it is clear that there is no intermediate or
extra phase formed in any of the composites synthesized. Hence there is no chemical
reaction between the two phases and it proves it is a composite of Ba(Zr,Ti)O3 and
CoFe2O4. No structural change is observed for both the phases in the composite. The
ferrite has the cubic spinel structure and Ba(Zr,Ti)O3 has perovskite structure.
The peaks of the perovskite structure shifts to the lower angle and is only due to the
change in the Zr concentration. There is no shift in the peaks of ferrite component. The
main peaks (110) of perovskite and (311) of spinel components of composites are shown
in Figure 4.10. The (311) peak of the spinel CFO phase remains at the same position
while the (110) peak of perovskite BZTx phase is shift to the lower angle.
30 31 32 33 34 35 36
*(311)
(110)
Intensity (a.u)
Angle 2
θ
(degree)
[BZ40]65:[CFO]35
[BZ50]65:[CFO]35
[BZ60]65:[CFO]35
[BZ70]65:[CFO]35
Figure 4.10: Peak shift in the XRD pattern with the Zr content in the [BZTx]0.65:[CFO]0.35.
The lattice parameters of both perovskite and cubic phases are calculated using Eq.
4.8 and are given in the Table 4.5. In comparison to the lattice parameter of pure relaxor
ferroelectric (Table 4.3) and cobalt ferrite sample, no significant change observed in the
lattice constant.
Chapter 4 PROCESSING AND STRUCTURAL CHARACTERIZATION
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 69
Table 4.5: Values of the lattice parameter of [BZTx]0.65:[CFO]0.35 calculated from the XRD
diffractogram.
-x Formula Lattice Parameters (Å)
BZTx CFO
0.4 [BZT0.4]0.65:[CFO]0.35 4.082 ± 0.003 8.389 ± 0.002
0.5 [BZT0.5]0.65:[CFO]0.35 4.101 ± 0.003 8.387 ± 0.004
0.6 [BZT0.6]0.65:[CFO]0.35 4.117 ± 0.003 8.386 ± 0.001
0.7 [BZT0.7]0.65:[CFO]0.35 4.136 ± 0.001 8.384 ± 0.003
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 70
Chapter 5 RELAXOR FERROELECTRICS:
ZIRCONIUM DOPED BARIUM
TITANATE
This chapter deals with the dielectric properties of the BaZrxTi1-xO3 solid solution. We
report the relaxor behavior of the zirconium doped barium titanate BaZrxTi1-xO3 solid
solutions and discuss the temperature, frequency and concentration dependence in
terms of correlations among the polar nano regions. Different models are employed to
find the critical parameter of relaxor system. The relaxor behavior is analyzed within
the mean field theory by estimating the Edward-Anderson order parameter EA
q.
Additionally we find that the calculated EA
q for the different concentrations obeys a
scaling behavior,
()
n
mEA TTq /1−= where Tm are the respective dielectric maxima
temperatures and n = 2.0 ± 0.1. The frequency dependence of the order parameter also
shows results consistent with the above mentioned picture.
Barium titanate, BaTiO3, is one of the most widely used ferroelectric materials for
capacitor applications. Pure barium titanate however, strongly suffers from temperature
dependent capacitance due to its sharp phase transition temperatures. This sharp phase
transition and the existence of the ferroelectric phase at room temperature limits the
usefulness of the pure material in practical applications [158]. It is well known that
BaTiO3 undergoes three successive structural transitions with decreasing temperatures.
These are cubic to tetragonal, tetragonal to orthorhombic and finally orthorhombic to
rhombohedral. The nature of pure phase transition temperatures and dielectric
characteristics of BaTiO3 are strongly dependent on the stoichiometry and composition
and therefore can be modified by partial substitution of minor amounts of foreign ions
into either the A-site (Ba-ion) or the B-site (Ti-ion) [46, 124]. BaTiO3 itself does not have
ideal properties for some industrial applications. Chemical substitutions at Ba2+ or Ti4+
sites can significantly change the behavior of BaTiO3 to meet a variety of device and
performance requirements [124, 159, 160].
Barium Zirconate Titanate (BaZrxTi1-xO3) is the most studied solid solution for
dielectric applications with zirconium substituted for titanium in the barium titanate
perovskite structure [95, 96]. BaZrxTi1-xO3 is interesting because its parent compounds are
rather different: BaZrO3 is paraelectric while BaTiO3 is a typical ferroelectric and the
solid solution itself forms a relaxor over a wide range of composition. Due to its well-
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 71
studied nature, from various dielectric aspects, this system forms a natural base from
which to address some of the fundamental issues and questions that arise in the context of
relaxors and their often complex, glass-like properties.
The detailed picture of relaxors has been reviewed in Chapter 1; here we recall
some of the outstanding major questions. Firstly, it is common to describe the relaxors as
consisting of polar nano regions with ordered dipoles within the regions and randomized
interactions between the various regions. These interactions between the PNRs are
understood to be at the heart of the relaxor properties such as frequency dependence of
the peak temperature etc. This obvious similarity to their magnetic analogues viz. spin
glasses has led to a variety of theories that borrow from the spin glass description in terms
of an order parameter. In this chapter we will study the temperature and frequency
dependence of the ferroelectric response and attempt to present the results in terms of an
order parameter variation with the said variables.
We will present the dielectric properties of the ceramic compositions with general
formula BaZrxTi1-xO3 (with 0 ≤ x ≤ 0.8). The synthesis part already been described in
chapter 4. Our emphases here will be on the relaxor behavior of the zirconium doped
barium titanate BaZrxTi1-xO3 solid solutions and we will discuss the temperature,
frequency and concentration dependence in terms of correlations among the polar nano
regions (discussed at length in chapter 1). Different models have been employed to find
the critical parameters of relaxor system. We argue that this concept provides a consistent
picture describing various anomalous features of relaxors.
The experimental part is as follows:
5.1 Dielectric Spectroscopy
The dielectric measurements were carried out for the ceramic compositions
BaZrxTi1-xO3 (0 ≤ x ≤ 0.8) at and below the room temperature with different applied
frequencies and using a four probe system (see Chapter 3). Both the capacitance C and
dissipation factor tanδ measurements were carried out over a frequency range 0.2 to 500
kHz using Wayne Kerr LCR meter (WK-4275) in the temperature range of 10-300 K
using a helium closed cycle system (Janis CCS-350 Cryostat) and high temperature
dielectric measurement setup. Before taking the measurements, calibration of setup was
done. Dielectric measurements have been taken using principle of parallel plate capacitor.
The disc shaped pellets having diameter 10-12 mm and thickness 0.6-1.0 mm with
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 72
electrode on both side are used for the measurements. The dielectric constant
(permittivity) of samples was calculated by using the relations
A
Cd
o
ε
ε
=
′ and
δ
ε
ε
tan
′
=
′′ 5.1
5.1.1 Temperature Dependent Dielectric Spectra of BaTiO3
The temperature dependence of real and imaginary part of the dielectric permittivity
(ε = εʹ+iε″) of BaTiO3 sample measured at 500 kHz, and at applied AC field of 50 mV is
shown in the Figure 5.1.
0 100 200 300 400 500
0
2000
4000
6000
8000 TC
T1
T2
ε
'
ε
"
Temperature (K)
ε
'
0
500
1000
1500
2000
ε
"
Figure 5.1: Dielectric spectrum of BaTiO3 as a function of temperature measured at 500 kHz
frequency.
From the figure it is clear that with decreasing temperature the dielectric spectrum
(real part εʹ) shows all the three transitions which is well supported by earlier results [26],
the first being a very sharp transition is the paraelectric to ferroelectric phase transition in
which the structure changes from cubic to tetragonal at temperature TC = 393 K. The
second is a ferroelectric to another ferroelectric transition in which the structure changes
from tetragonal to orthorhombic at T1=278 K and the third is again from ferroelectric to
ferroelectric at T2=183 K and structure changes from orthorhombic to rhombohedral [26,
99]. The imaginary part ε″ also shows peaks corresponding to these transitions. The
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 73
values of the εʹ and ε″ at the transition temperature TC are 6883 and 670 respectively. The
reason of the ferroelectric behavior in BaTiO3 is very well known and is due to the mode
softening at the transition temperature [26].
It is well known from literature that in normal ferroelectric materials the dielectric
permittivity above the transition temperature obeys the Curie-Weiss law (CW) for electric
susceptibility as given in Eq. 5.2
C
T
θ
ε
−
=
′
1 )( C
TT > 5.2
Here θ is the Curie temperature; C is the Curie-Weiss constant and TC is the Curie point
or transition temperature. The fitted data taken at 500 kHz frequency are shown in Figure
5.2 that yields the values of Curie constant C and Curie temperature θ.
300 350 400 450 500
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
TC
1/
ε
'
Temperature (K)
θ
Figure 5.2: The inverse of the dielectric permittivity (1/εʹ) as function of temperature of BaTiO3
sample for f = 500 kHz. The line shows the fit to the Curie-Weiss behavior.
It is found that the data completely follow the Curie Weiss law in the temperature
range above the transition temperature TC. The values of the critical parameters C and θ
obtained from the fitted data are 1.28×105 and 377 K respectively. Since the value of θ <
TC, the transition from paraelectric to ferroelectric is of first order in BaTiO3 [14].
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 74
5.1.2 Temperature Dependence of Dielectric Spectra of BaZr0.3Ti0.7O3
The temperature dependence of real and imaginary part of the dielectric permittivity
of BaZr0.3Ti0.7O3 sample measured at 500 kHz, and at applied AC field of 50 mV is shown
in the Figure 5.3.
0 50 100 150 200 250 300
0
1000
2000
3000
4000
5000
ε
'
ε
"
Temperature (K)
ε
'
0
100
200
300
400
500
600
ε
"
Figure 5.3: Dielectric spectrum of BaZr0.3Ti0.7O3 as a function of temperature.
It is clear from the figure that with decreasing temperature the real component of the
permittivity εʹ increases gradually, passes through broad maximum and then smoothly
decreases with decreasing temperature. The broad maximum in εʹ is an important
characteristic of the disordered perovskite structure with a diffuse phase transition (DPT).
The imaginary part ε″ also shows broad peak corresponding to this transition. It is evident
from Figure 5.3 that the dielectric peaks of both real and imaginary components of the
dielectric susceptibility are at somewhat different temperatures. For example, the peak
position of the real part of dielectric constant occurs at the temperature Tʹm (= 228 K),
whereas the peak position of the imaginary part occurs at Tʺm (= 198 K). This non
equality of peak temperature obtained from εʹ(T) and ε″(T) curves has been argued to be
an indication of the existence of the DPT [8].
Similar measurements were also performed for other composition and for different
measuring frequencies and will be discussed later in this section.
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 75
5.1.3 Reduced Dielectric Spectra of BaZrxTi1-xO3
In Figure 5.4, both real and imaginary parts of the dielectric susceptibility of
BaZrxTi1-xO3 are shown for different Zr concentrations. The dielectric susceptibility has
been normalized with respect to its peak value for comparison.
Figure 5.4: Reduced dielectric spectra, a) real part, b) imaginary part of dielectric permittivity for
BaZrxTi1-xO3 with (0.3 ≤ x ≤ 0.8).
0 50 100 150 200 250 300
0.0
0.2
0.4
0.6
0.8
1.0
(b)
ε
"/
ε
m
"
Temperature (K)
x = 0.3
x = 0.4
x = 0.5
x = 0.6
x = 0.7
x = 0.8
0.0
0.2
0.4
0.6
0.8
1.0
(a)
ε
'/
ε
m
'
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 76
It is clear from the figure that the dielectric peak shifts to low temperature as the Zr
concentration increases. Peak broadening with increasing Zr concentration is also evident.
The broad peaks in all of these compositions represent diffuse phase transition in these
systems. This has been observed in other similar systems and may be attributed to
inhomogeneous distribution of Zr ions in the Ti-sites and/or to mechanical stress in the
grain. It is expected that the disorder in the cation distribution causes the DPT where the
local Curie points of different nano regions are statistically distributed in a wide
temperature range around the mean Curie temperature [40, 161-163]. Another possible
explanation of this broad maximum is in the light of superparaelectric model. According
to this model the dielectric permittivity of the relaxor is related to the thermally activated
reorientation of the local spontaneous polarization in polar nano regions distributed in the
non-polar matrix. At high temperatures, PNRs have the size of several nanometers and
are highly dynamic. However on cooling further the interactions between the PNRs starts
and become stronger with further decrease in temperature. As a consequence the slowing
down of their dynamics occurs (i.e. thermal disordering effect start to decrease). A broad
distribution of PNR sizes and randomness of interaction between them lead to a broad
distribution of relaxation time giving rise to a broad peak of εʹ. It is also evident from the
in-phase part data that below the main peak a shoulder is developing in the dielectric
response at lower temperatures, for the highest Zr concentrations. This is most clearly
manifested in the x = 0.8 composition, where a clear slope change is observed at 66 K
before reaching the maximum at 126 K. For this concentration the out of phase part
shows the clear emergence of two peaks suggesting a change in the loss mechanisms
possibly due to change in the relaxation mechanisms at the lower temperatures. This
composition lies in the polar cluster region of the phase diagram (Figure 1.14).
The physical picture that has emerged is as follows. It is clear that chemical
substitution and lattice defects can introduce dipolar entities in mixed ABO3 perovskites.
At very high temperatures, thermal fluctuations are so large that there are no well-defined
dipole moments. However, on cooling, the presence of these dipolar entities manifests
itself at a temperature (the so-called Bums or dipolar temperature) TB > Tm. At and below
TB each dipolar entity will induce polarization (or dipoles) in adjoining unit cells of its
highly polarizable host lattice, forming a dynamic polarization "cloud" (Figure 5.5)
whose extent is determined by the polarizability, or correlation length for dipolar
fluctuations, rc. Near TB, rc is small and the polarization clouds are effectively small polar
Ch
a
S
tu
d
nan
inc
r
nan
and
b
ut
do
w
ori
e
Fig
u
tem
p
gro
w
5.6.
inc
r
con
of
T
alo
n
tetr
a
ph
o
tha
t
the
r
b
o
n
the
r
sub
s
a
pter 5
RE
d
ies of Ferro
e
odomains
r
eases in
a
odomains,
Coulombi
c
do not be
c
w
n of their
e
ntation of
t
u
re 5.5: Ra
n
p
erature (T
m
w
and coales
c
The pea
k
It is clear
f
r
easing
Zr
c
centrations
T
ʹ
m
and
Tʺ
m
In a prop
n
gside a s
o
a
gonal) an
d
o
non mode
i
t
ferroelectr
r
efore play
a
n
ds at the s
u
r
efore be
r
s
tituent.
E
LAXOR F
E
e
lectric and
M
(Figure 5.
5
a
Curie-W
e
coupling t
h
c
interactio
n
c
ome large
fluctuation
s
t
he pola
r
do
n
domly dist
r
< T < T
B
) t
h
c
e at low tem
p
k
temperat
u
f
rom the fi
g
c
oncentrati
o
is evident
i
is maximu
m
er ferroele
c
o
ft mode
d
d
conseque
n
i
n this case
icity in the
a
key role i
n
u
bstituted si
r
esponsibl
e
E
RROELE
C
M
ultiferroic B
e
5
(a)
). Ho
w
e
iss-like m
a
h
em into gr
o
n
s (Figure
enough, t
h
s
at
T < T
m
mains.
r
ibuted pola
r
h
e polar nan
o
p
eratures.
u
res
T
m
as
a
g
ure that bo
o
n. The in
e
i
n all the
BZ
m
in the ra
n
c
tric the ph
a
d
istortion t
h
n
t develop
m
refers to t
h
case of BT
n
the ferro
e
tes are exp
e
e
for vary
i
C
TRIC
S
: ZI
R
e
havior in [B
a
w
ever, wit
h
a
nner, rapi
o
wing pola
r
5.5
(b)
). T
h
h
en they w
i
m
, leading
t
r
nano regi
o
o
regions an
d
a
function
o
th the peak
e
quality of
Z
Tx
compo
s
n
ge (
0.45 ≤
x
a
se transitio
h
at leads
t
m
ent of a
d
h
e Ti-O bo
n
O arises fr
o
e
lectricity o
e
cted to be
i
ng bond
R
CONIUM
D
a
(Zr,Ti)O
3
]
1-
y
h
decreasi
n
dly increa
s
r
cluster a
n
h
e nanodo
m
i
ll ultimate
t
o an isotr
o
o
ns in a so
f
d
the correl
a
o
f Zr conc
e
temperatu
r
Tʹ
m
and
Tʺ
m
s
itions. The
x
≤ 0.65
).
n from par
a
t
o a struct
u
d
ipole mom
e
n
d and its
o
o
m the dist
o
f BTO. On
strongly af
f
strengths,
D
OPED
BA
y
:[CoFe
2
O
4
]
y
S
n
g temper
a
s
ing
r
, an
d
n
d increasi
n
m
ains grow
ly exhibit
a
pic relaxor
f
t-mode hos
t
a
tion radius,
e
ntration a
r
r
es (
Tʹ
m
and
m
noted pr
e
difference
a
electric to
u
ral instab
i
e
nt in eac
h
o
scillations
o
rtion of th
e
substitutin
g
f
ected and
t
at rando
m
A
RIUM TIT
A
S
ystem
a
ture of
t
d
the size
s
n
g their cor
r
with decre
a
dynamic
state with
t
lattice. (a)
r
c
are small.
r
e plotted i
n
Tʺ
m
) decre
a
e
viously fo
r
between t
h
ferroelectri
i
lity (e.g.
c
h
unit cell.
T
[14, 164].
W
e
Ti-O bon
d
g
Zr for Ti
t
t
he substitu
t
m
positions
A
NATE
| 77
t
he host
s
of the
r
elations
asing
T,
slowing
random
At high
(b) Both
n
Figure
ase with
r
low Zr
h
e values
c occurs
c
ubic to
T
he soft
W
e note
d
s which
t
he Ti-O
t
ion will
of the
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 78
0.3 0.4 0.5 0.6 0.7 0.8
90
120
150
180
210
240
T m (K)
Zr Conc. (x)
T'm
T"m
Figure 5.6: Variation in peak temperature of real and imaginary part of ε′ of with Zr concentration.
The decrease of the peak temperature with the Zr concentration from paraelectric to
ferroelectric with diffuse phase transition is understood to be due to the larger ionic radius
of Zr4+ than that of Ti4+ ion. Substitution of Zr4+ for Ti4+ results in weakening of bonding
force between the B-site ion and oxygen ion of the ABO3 perovskite structure. As the B-O
bonds are weakened, the B-site ion can resume its position only when the cubic
paraelectric to tetragonal ferroelectric phase transition is at lower temperature [165].
The variation in the values of εʹm and εʺm with the Zr concentration found from the
data taken at 500 kHz is shown in the Figure 5.7. It is clear that the absolute values of
both the εʹm and εʺm decrease with increase in the Zr concentration. The dielectric peak
gets suppressed as we increase the Zr content which might be understood as the
substitution of Zr4+ in polar BaTiO3 matrix that breaks the correlations in displacement of
Ti4+ cations and related long range polar order.
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 79
0.0 0.2 0.4 0.6 0.8
0
1000
2000
3000
4000
5000
6000
7000
8000
Zr Conc. (x)
ε
'm
-100
0
100
200
300
400
500
600
700
ε
"m
Figure 5.7: Variation in absolute value of ε′m and εʺm with Zr concentration measured at 500 kHz.
The loss peak also gets suppressed upon 30% Zr content in the host compound
BTO. This suppuration increases with increase in the Zr content. This can be explained as
being due to the suppression of hopping transport between adjacent Ti ions (Ti3+ and Ti4+)
on Zr addition. Hopping transport or conduction is one of the main reasons for enhanced
dissipation in the parent system. While Ti is present initially as both Ti+3 and Ti+4 in the
parent system, the valance of Zr4+ being chemically more stable results on predomination
of Ti+4 on substitution. Thus in the Zr doped system the opportunities for hopping
between trivalent and tetravalent ions is reduced leading to lowered dissipation.
Furthermore the large ionic radius of Zr4+ ion expands the perovskite lattice and enlarge
the hopping distance. Therefore conduction by electron hopping between the adjacent Ti
can be depressed by the substitution of Ti with Zr, resulting in poor conduction [165,
166]. Thus the εʺm peaks reflecting the conductivity related losses decrease due to
decrease in the ionic conductivity as Zr content increases. Similar results on decrease of
loss factor with increasing Zr were also observed at 0.2 kHz, 1 kHz, 10 kHz, and 100 kHz
frequency of applied field (see section 5.1.5).
Diffuse phase transition is not a sufficient characteristic for the relaxor behavior;
another characteristic is the frequency dependence of the dielectric permittivity peak
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 80
temperatures (Tʹm and Tʺm). It has been shown that above TC ferroelectric materials follow
the well-known Curie Weiss law (as discussed in chapter 1). Furthermore it has been
reported that ferroelectric relaxors show a deviation from the Curie-Weiss behavior in the
vicinity of peak temperature and obey the so called modified Curie-Weiss law in the high
temperature paraelectric phase above peak temperature. In the subsequent section we
investigate these characteristic of the prepared BZTx compositions.
5.1.4 Dielectric Spectra of BaZrxTi1-xO3 and Curie Weiss Behavior
In normal ferroelectric materials the dielectric permittivity above the transition
temperature obeys the Curie-Weiss law (CW) for electric susceptibility as given in Eq.
5.2. However, the dielectric permittivity εʹ of the relaxor ferroelectric above the peak
temperature (in the high temperature paraelectric phase) is not completely described by
the Curie Weiss law (Eq. 5.2) and a strong deviation is observed above the peak
temperature. To determine the critical parameters C (Curie constant) and the Curie
temperature θ, the higher temperature data of the dielectric constant
ε
ʹ was fitted to Eq.
5.2 for all the BZTx compositions. Note that the values of these critical parameters
strongly dependent on the temperature range being fitted [130].
The fitted data taken at the 500 kHz frequency are shown in Figure 5.8 that yields
the values of Curie constant C and Curie temperature θ. The values of the critical
parameters obtained from the fitted data are shown in the Table 5.1. These values are
consistent with those reported previously [9].
The deviation of the experimental data from the linear curve is clear from the Figure
5.8. Note that the deviation of the data from the Curie-Weiss behavior initiates at a
temperatures Tdev higher than the peak temperature Tm, as marked in Figure 5.8 for the
BZT0.6. From the Table 5.1, it is clear that the value of θ decreases with increasing Zr
content. The decrease in θ is understood to be due to the larger ionic radius of Zr+4 (0.72
Å) than that of Ti+4 (0.605 Å) [167]. The oriented displacement of the B-site ions in the
oxygen octahedra is understood to lead to the development of ferroelectric correlations.
However because of the larger radius of Zr4+ its substitution for Ti4+ is expected to
suppress this oriented displacement of the B-site ions in the oxygen octahedra. Therefore
the interactions between the B-site ions and O2− is expected to become weaker, resulting
in a decrease in phase-transition temperature as well as the curie temperature θ [168,
169].
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 81
0 50 100 150 200 250 300
0.000
0.005
0.010
0.015
0.020
0.025
0.030
θ
(BZT0.6)
Tdev(BZT0.6)
1/
ε
'
Temperature (K)
BZT0.3
BZT0.4
BZT0.5
BZT0.6
BZT0.7
BZT0.8
Figure 5.8: The inverse of the dielectric permittivity (1/εʹ) as function of temperature for various
concentrations, for f = 500 kHz. The lines are the fit to the Curie-Weiss behavior. The deviation
temperature Tdev and the x-intercept θ are marked for the BZT0.6.
Table 5.1: The Critical parameter (Curie-Weiss constant C, Curie-Weiss temperature θ, the deviation
temperature Tdev, peak temperature Tm and the degree of deviation ΔTm) calculated from the Curie-
Weiss law fit of the experimental data in the high temperature paraelectric regime.
Sample C θ (K) Tde
v
(K) Tm (K) ΔTm (K)
BaZr0.3Ti0.7O3 1.11×105 228.10 ± 2 261 237 24
BaZr0.4Ti0.6O3 5.0×104 161.5 ± 1 242 172 70
BaZr0.5Ti0.5O3 3.33×104 91 ± 2 234 146 88
BaZr0.6Ti0.4O3 2.5×104 25.75 ± 2 207 152 55
BaZr0.7Ti0.3O3 3.33×104 -84.67 ± 1 186 144 42
BaZr0.8Ti0.2O3 2.0×104 -187.6 ± 2 180 126 54
The negative value of θ at higher concentrations of Zr is also evident from the table.
The compositions with negative values of θ suggest the presence of anti-ferroelectric
interactions in the system [170, 171]. It has also been suggested that this may arise due to
some secondary influential features (such as quantum fluctuations etc.), near and below
the relaxation peak [9, 100]. (It will be seen in our data to be shown that the dynamical
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 82
behavior in the compositions with negative value of θ is rather different.) However, it is
seen that the dielectric permittivity of all the BZTx compositions follows Curie–Weiss law
at temperatures much higher than the Tʹm. This deviation from the Curie Weiss behavior is
typically an indication of interaction between the dipoles or polar nanoregions. These
interactions can start well above the peak temperature and may lead to a blocking or
freezing of the moments. In the next section we will explain this deviation in terms of
correlation among the polar nano regions by calculating Edwards Anderson order
parameter.
To illustrate the degree of deviation from the Curie-Weiss law, the parameter ΔTm,
can be defined as [172]
mdevm TTT −=Δ 5.3
It is observed from the data in the Table 5.1, the value of ΔTm increases with
increase in the Zr content and passes through a maximum value for the BZT0.5 and
afterward it decreases.
It is worth mentioning here that for the relaxor ferroelectrics that show DPT, the
complete fit to data from high temperature up to peak temperature Tʹm, is usually
expressed by the modified Curie–Weiss Law [41, 71, 173-175]. The equation for
modified Curie-Weiss law is given in Eq.1.10.
By solving Eq. 1.10, we get
()
1
lnln
11
ln CTT m
m
−−=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
′
−
′
γ
εε
5.4
It is an equation of straight line, with
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
′
−
′
=
m
y
εε
11
ln and
()
m
TTx −= ln 5.5
The slope of the line gives the value of γ and from the intercept we calculate the
value of C1. Equation 5.4 can be solved graphically using log-log plot i.e.
()
m
ε
ε
′
−
′/1/1ln
vs.
()
m
TT ′
−ln as shown in Figure 5.9. Scatter points show the experimental data and line
on the graph represents the modified Curie-Weiss law fit.
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 83
Figure 5.9: Plot of ln(1/εʹ-1/εʹm) vs. ln(T-Tʹm) of the data taken at 500 kHz applied frequency. The lines
are the fit to the Modified Curie-Weiss law (Eq. 5.5). The values of the slope γ are mention for all the
BZTx sample.
Table 5.2: The critical parameters (Modified Curie-Weiss constant C1 and the diffuseness exponent γ)
calculated from the Modified Curie-Weiss law fit of the experimental data in the temperature range
above Tʹm.
Formula C1 -γ
BaZr0.3Ti0.7O3 9.07×106 1.98 ± 0.01
BaZr0.4Ti0.6O3 4.87×106 1.88 ± 0.02
BaZr0.5Ti0.5O3 1.42×106 1.67 ± 0.01
BaZr0.6Ti0.4O3 6.81×106 1.65 ± 0.02
BaZr0.7Ti0.3O3 6.0×106 1.60 ± 0.02
BaZr0.8Ti0.2O3 1.58×106 1.84 ± 0.01
The values of the fitting parameters are listed in Table 5.2 for the data taken at 500
kHz. The values of γ lie between 1 and 2. The given values of γ are evidence to the
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
-12
-10
-8
-6
-4
γ
= 1.84
ln(T-T'm)
BZT0.8
MCW Law fit to Data
2.53.03.54.04.55.0
-10
-9
-8
-7
-6
-5
γ
= 1.60
ln(1/
ε
'-1/
ε
'
m
)
ln(T-T'm)
BZ7T0.7
MCW Law fit to Data
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
-12
-10
-8
-6
-4
γ
= 1.65
ln(T-T'm)
BZT0.6
MCW Law fit to Data
0.51.01.52.02.53.03.54.04.55.05.5
-14
-12
-10
-8
-6
-4
γ
= 1.67
ln(1/
ε
'-1/
ε
'
m
)
ln(T-T'm)
BZT0.5
MCW Law fit to Data
2.02.53.03.54.04.55.0
-12
-10
-8
-6
-4
γ
= 1.88
ln(T-T'm)
BZT0.4
MCW Law fit to Data
1.01.52.02.53.03.54.04.5
-14
-13
-12
-11
-10
-9
-8
-7
γ
= 1.98
ln(1/
ε
'-1/
ε
'
m
)
ln(T-T'm)
BZT0.3
MCW Law fit to Data
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 84
diffuse phase transition happened in the BZTx composition. The values of γ are found to
vary from 1.60 to 1.98. These values support the presence of relaxor nature of the BZTx
ceramics.
5.1.5 Frequency Dispersion and Freezing Behavior in BaZrxTi1-xO3
To study the frequency dispersion around the transition temperature in the BZTx
sample, we measure the dielectric constant as function of temperature at different
frequencies.
0 100 200 300 400 500
0
2000
4000
6000
8000
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
Temperature (K)
ε
'
0
500
1000
1500
2000
2500
T2
ε
"
TC
T1
Figure 5.10: Frequency dependence of dielectric spectra of BaTiO3 ceramic samples.
Figure 5.10 shows the temperature dependence of εʹ and εʺ as function of
temperature for BaTiO3 ferroelectric sample measured at five different frequencies. It is
clear from the figure that no frequency dispersion is observed at and around all the three
transition temperatures. The absence of frequency dispersion is one of the characteristics
of the normal ferroelectrics. Large frequency dispersion of the dielectric properties at low
temperature has been considered as the characteristic property of relaxor ferroelectrics.
The temperature dependence of both the real and imaginary part of dielectric
susceptibility of Ba(Zr,Ti)O3 composition with 0.3 ≤ x ≤ 0.8 at various frequencies are
shown in Figure 5.11 (a-f).
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 85
Figure 5.11: Frequency dependence of dielectric spectra of BZTx ceramic samples.
As expected, dielectric measurements show frequency dispersion at the temperature
of the maximum of the dielectric curves allowing characterizing the samples as relaxor
ferroelectric materials. It is seen in all the BZTx compositions that, with increasing
frequency the maximum value of real part of the dielectric susceptibility (εʹm) decreases,
whereas the maximum value of the imaginary part (εʺm) increases. The peak temperatures
where the dielectric maxima occur for both the real and imaginary part (Tʹm and Tʺm) are
shifted to higher temperatures with the increase in the frequency.
0 50 100 150 200 250 300
30
40
50
60
70
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
(f)BZT0.8
Temperature (K)
ε
'
0
2
4
6
8
ε
"
0 50 100 150 200 250 300
40
60
80
100
120
140
(e)BZT0.7
Temperature (K)
ε
'
0
5
10
15
20
25
30
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
ε
"
0
50
100
150
200
250
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
(d)BZT0.6
ε
'
0
10
20
30
40
50
60
70
ε
"
0
100
200
300
400
500
600
(c)BZT0.5
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
ε
'
0
50
100
150
200
250
ε
"
0
500
1000
1500
2000
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
(b)BZT0.4
ε
'
0
200
400
600
800
1000
ε
"
0
1000
2000
3000
4000
5000
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
ε
'
(a)BZT0.3
0
200
400
600
800
1000
ε
"
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 86
It is the characteristic of relaxor that their dielectric properties are controlled by a
broad spectrum of relaxation times [72]. This distribution of relaxation times is
understood to be brought about by the distribution of the sizes of the polar nanoregions.
The DPT is regarded as being the outcome of the local composition fluctuations caused
by the polar TiO6 and non-polar ZrO6 resulting in nanoregions with different local Curie
temperatures (transition temperatures) distributed in a Gaussian fashion around a mean
Curie temperature.
The properties of relaxors are closely related to their unique polar structure, namely
to the existence of polar clusters of nanometer-size, each of which has a net dipole
moment. These polar clusters are called polar nano regions (PNRs). These PNRs are
considered as separated regions of nanometer size possessing a spontaneous polarization
(PS) and having a characteristic relaxation time controlled by the local field configuration.
The dielectric response is interpreted as a result of reorientation of the local polarization
vectors under the applied electric filed [43]. The dipole moments have different stable
energy minima, which are separated from others by an energy barrier with height of the
order of the thermal energy (25 meV) or a little higher. In presence of any ac field, the
observed dielectric properties would correspond to the cumulative response of all polar
clusters. The number of polar clusters responding to the external field depends on several
factors such as: (a) temperature, (b) amplitude of the ac signal and (c) frequency of the ac
signal, and/or any defects sites. As the temperature increases, the height of the energy
barrier between the different minima is reduced, which in turn enhances the number of
responding polar entities.
5.1.6 Relaxation Time and Vogel-Fulcher Law
The superparaelectric model, the Vogel Fulcher dipole glass model and dipolar
dielectric model are all based on the assumption that the mechanism of the response
function (dielectric constant) is related to the thermally activated reorientation of the local
spontaneous polarization in polar nano regions distributed in the non-polar matrix. At
high temperatures it is assumed that the PNRs act as noninteracting entities and these
PNRs start interacting as the temperature is lowered. However on cooling below Tdev the
interactions between the PNRs become stronger and as a consequence the slowing down
of their dynamics occurs (i.e. thermal disordering effect start to decrease) and the number
density and volume fraction of the polar nano regions increase. Each polar nanoregion has
a net polarization (PS) and has a characteristic relaxation time (τ) that depends on their
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 87
characteristic size and randomness of interaction between them. A broad distribution of
PNR sizes and randomness of interaction between them lead to a broad distribution of
relaxation time giving rise to a broad peak of εʹ.
It is believed that short range interaction between PNRs controls the fluctuations of
polarization, PS, this leads to the freezing of PNRs at a characteristic temperature (TVF)
and hence the transition into the glass likes state. To elaborate this we have plotted ln(f)
vs. Tʹm and Tʺm for all the compositions in the Figure 5.12. Note that the obtained data
cannot be fitted with the simple Debye equation (Arrhenius law).
⎥
⎦
⎤
⎢
⎣
⎡−=
mB
a
Tk
E
ff exp
ο
5.6
Here f is the applied frequency while f˳ is a characteristic attempt frequency. In order to
analyze the relaxation features the experimental curves were fitted using the Vogel-
Fulcher law (Eq. 1.11) [176]. Figure 5.12 also show the fitting of data to Vogel- Fulcher
law Eq. 1.11. The fitting parameters (Ea activation energy, f˳ is a characteristic attempt
frequency and Vogel Fulcher freezing temperature TVF) obtained are given in Table 5.3
and Table 5.4.
The empirical relaxation strength ΔTrelax describing the frequency dispersion of Tm
is defined as [177]
)2.0()500( kHzmkHzmrelax TTT −=Δ 5.7
The values of empirical relaxation strength calculated from the data are also listed
in Table 5.3 and Table 5.4.
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 88
Figure 5.12: Peak temperature dependence of frequency for BZTx (0.2 ≤ x ≤ 0.8) ceramics. Solid lines
represent the fitting by using the Vogel-Fulcher equation 5.8.
60 70 80 90 100 110 120 130
(f) BZT0.8
T'
m
T"
m
VF law Fit to Data
T
m
(K)
60 80 100 120 140 160
4
6
8
10
12
14
(e) BZT0.7
ln (f)
T
m
(K)
60 80 100 120 140 160
(d) BZT0.6
T
m
(K)
70 80 90 100 110 120 130 140 150
4
6
8
10
12
14
(c) BZT0.5
ln (f)
T
m
(K)
80 100 120 140 160 180
(b) BZT0.4
T
m
(K)
190 200 210 220 230 240
4
6
8
10
12
14
T
m
(K)
(a) BZT0.3
ln (f)
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 89
Table 5.3: The critical parameters, pre-exponential factor fₒ, the relaxation time τₒ, activation energy
Ea and static freezing temperature TVF calculated from the Vogel-Fulcher law (Eq. 1.11) fit to the
experimental data (peak temperature of real part of dielectric constant Tʹm).
Composition fₒ (Hz) τₒ (sec) Ea (meV) TʹVF (K) ΔTʹrelax (K)
BaZr0.3Ti0.7O3 1.36×1010 7.30×10-11 17.80 216.63 9
BaZr0.4Ti0.6O3 1.29×1010 7.72×10-10 79.25 83.158 38
BaZr0.5Ti0.5O3 8.40×109 1.19×10-10 94.19 33.614 50
BaZr0.6Ti0.4O3 8.51×109 1.17×10-10 113.36 19.288 58
BaZr0.7Ti0.3O3 7.01×1013 1.42×10-14 211.93 13.709 38
BaZr0.8Ti0.2O3 2.11×1014 4.73×10-15 233.15 8.4581 48
Table 5.4: The critical parameters, pre-exponential factor fₒ, the relaxation time τₒ, activation energy
Ea and static freezing temperature TʺVF calculated from the Vogel-Fulcher law (Eq. 1.11) fit to the
experimental data (peak temperature of imaginary part of dielectric constant Tʺm).
Composition fₒ (Hz) τₒ (sec) Ea (meV) TʺVF (K) ΔTʺrelax (K)
BaZr0.3Ti0.7O3 8.40×107 1.18×10-8 18.17 181.78 22
BaZr0.4Ti0.6O3 1.61×108 6.22×10-9 34.14 74.88 40
BaZr0.5Ti0.5O3 1.05×1011 9.55×10-12 101.05 15.80 38
BaZr0.6Ti0.4O3 2.29×1010 4.35×10-11 91.73 13.12 40
BaZr0.7Ti0.3O3 1.19×1012 8.41×10-13 134.58 7.52 38
BaZr0.8Ti0.2O3 2.76×1012 3.62×10-13 125.61 3.27 30
The examination of our data from Table 5.3 and Table 5.4 shows that the value of
TʹVF > T
ʺVF for all the samples. It is clearly seen that with increase in Zr4+ content the
freezing temperature (TVF) is decreased systematically, whereas the activation energy
increases with increase in Zr4+ content. The systematically decreasing values of TVF
indicate the gradual evolution of relaxor behavior with the decrease in content of polar
BaTiO3 in nonpolar BaZrO3 matrix. It is suggested with the gradual incorporation of polar
BaTiO3 in the nonpolar dielectric BaZrO3 matrix, BZTx ceramics start showing polar-
cluster like behavior from a simple dielectric (for pure BaZrO3) and then probably a
critical size and density of the polar regions are reached when it starts showing relaxor
behavior. The values of the Debye frequencies fₒ or the relaxation time τₒ are in good
agreement with the previously reported data on these compositions [9].
The transformation to relaxor behavior in BZTx is related to the disorder within the
B-site of the perovskite-type ABO3 unit cell. Indeed, the ferroelectricity in barium titanate
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 90
resides on a cooperative shift of Ti4+ cation into a certain direction form the center of the
oxygen octahedron. The lager Zr cation cannot go off-center, giving rise to random
breaking of the correlated displacement along Ti-O-Ti-O chain. Spatial fluctuations of the
defected (broken) bonds result in fluctuations of the polar correlations which in turn are
responsible for the formation of precursor polar clusters already above Tʹm. As Zr content
increases, the regions with an accumulated concentration of broken bonds will occupy a
larger part of the sample. As a consequence the polar correlations are strongly diminished
and ferroelectric domains are less likely to nucleate. However due to the distortions
arising around the Zr ions a redistribution of the charges and a local formation of charged
centers results. These are sources of local random fields, whose quenched spatial
fluctuations acts as pinning centers of the thermally fluctuating polarization. Clearly, this
kind of random field is much weaker than that stemming from heterovalent cation
substitution as in conventional relaxors. Hence, the relaxor properties of BZTx require
relatively high doping level [8, 125].
5.2 Edward-Anderson Order Parameter and Scaling Behavior in
BaZrxTi1-xO3
It has been argued in the past and also in recent works that ferroelectric relaxors
have many features that resemble those of magnetic spin glass systems and are
describable generally by the models developed for those systems [1]. Akbarzadeh et al in
their numerical studies have described the behavior of the dielectric susceptibility for
BaZr0.5Ti0.5O3 [129], using the Edward-Anderson order parameter [178]. Their work also
illustrates the presence of polarization clusters that grow in size and number with
decreasing temperatures and begin to interact below a certain characteristic temperature.
These polarization clusters are identical to the polar nano regions (PNRs) that are
understood to be central to the relaxor behavior, with the addition that they are now
considered to be dynamic entities i.e. the center of the cluster may shift from one Ti-ion to
another with time. The predictions of this model were found to be in good agreement with
the experiments of Bhalla et al [9] for the temperature and frequency dependence of the
dielectric constant of BZT materials. This is consistent with the generally accepted
picture of the development of polar nano regions, their growth and the development of
interactions between them. However it remains to be seen if (and how) these features
manifest themselves experimentally in the order parameter, both with increasing levels of
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 91
substitution and increasing frequency. In this section we are going to analyze the electric
susceptibility data of BaZrxTi1-xO3 relaxor ferroelectrics (0.3 ≤ x ≤ 0.6) in the light of
mean field theory. We present the variations of the obtained Edward-Anderson (EA) order
parameter for these variables (x, T, f). We further explore the important question whether
the order parameter for different concentrations can be described by a universal behavior
)/( mEA TTq , as is typically seen in classical spin glasses. In a similar vein the variation of
),( fTq EA is studied to determine how the dynamical response at different temperatures
manifests itself in the order parameter.
It has been argued [1, 161] that relaxor ferroelectrics may be considered as electric
dipole analog of magnetic spin glass systems and therefore relaxor ferroelectrics may be
treated and analyzed using the well-established models such as the Edwards-Anderson
model developed for the spin glass systems. Hence to discuss the development of the
relaxor state we take into account the EA-order parameter which in the present context
describes the average correlations between the different PNRs. The order parameter EA
q
can be written as jiEA PPq ≡ [130, 179, 180], here Pi and Pj are the dipole moments
corresponding to the ith and jth polar nano regions respectively. Sherrington and
Kirkpatrick (SK) developed [127] an infinite range model for the spin glass to calculate
the EA-parameter. This model relates the temperature dependence of the susceptibility (χ)
to the local order parameter EA
q as given in Eq. 1.13[127, 130, 181, 182].
Vugmeister and Rabitz derived a relation similar to Eq. 1.13 for the relaxor
ferroelectric as discussed in chapter 1 (see section 1.8.3). Equation 1.23 has been used
extensively to extract the EA-order parameter from the experimental data for relaxors. By
solving Eq. 1.23 we get
()
C
CTT
TqEA +
′
+−
′
=
θε
θ
ω
ε
ω
),(
),( 5.8
We determine EA
qusing Eq. 5.8. For this purpose, we use the values of C and θ as
determined from the Curie Weiss fit of the data (Table 5.1), and the value of dielectric
susceptibility (at a particular frequency) for varying temperature (Figure 5.11 (a-d)) are
inserted in Eq. 5.8. The results for an applied frequency of 500 kHz are shown in Figure
5.13.
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 92
0 50 100 150 200 250 300
0.0
0.2
0.4
0.6
0.8
1.0
qEA
Temperature (K)
BZT0.3
BZT0.4
BZT0.5
BZT0.6
Tdev(BZT0.3)
Figure 5.13: The EA order parameter EA
q as a function of temperature for various BZTx
compositions and f = 500 kHz determined using Eq. 5.11. Note that the onset temperature Tdev of EA
q
is the same as the deviation temperature determined from the Curie-Weiss behavior
It is evident from the figure that for all concentrations of Zr,
()
TqEA starts smoothly
and slowly at high temperatures and then rises rapidly and appears to merge at low
enough temperatures. Interestingly for each composition the value of EA
q becomes
nonzero at a temperature closely coinciding with the temperature Tdev defined earlier as
the temperature where the 1/
ε
ʹ vs. T data begin to deviate from the Currie–Weiss fit (see
Figure 5.8). In general we observe that the onset point for nonzero EA
q value shifts to
lower temperatures with increasing x. Note that at fixed temperature e.g. at 100 K,
()
TqEA
for higher concentrations is lower than
()
TqEA at low concentrations. All these
observations are consistent with the PNRs or polar clusters picture [129]. For example at
low Zr concentration paraelectric BaZrO3 forms a dilute solution in ferroelectric BaTiO3.
At this stage, it is suggested that the polar clusters that are centered at Ti+4 sites are in
close proximity to each other and are able to develop strong correlations even at relatively
higher temperatures. On the other hand increasing Zr concentration results in a lower
number of Ti-centers in the BZT solid solution and hence to poor correlations among the
PNRs at higher temperatures. As the temperature is reduced these correlations therefore
grow rapidly with more and more PNRs being formed, while the smaller PNRs merge to
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 93
form even bigger clusters. Meanwhile the EA-order parameter
()
TqEA that reflects the
growth of these correlations also increases. Finally, at low enough temperatures the
number of PNRs begins to saturate and the order parameters
()
TqEA for the respective
concentrations merge at low enough T.
Our analysis of the data up to this point has shown that the evolution of the
dielectric response can be described in terms of a glass-like order parameter EA
q that rises
from zero at the point where the correlations manifested in the deviation from the CW
law originate and then saturates to the full value at low temperatures. However for an
order parameter to be taken seriously it must, in general, exhibit a general or universal
behavior as the function of scaled temperature. In order to determine if the behavior of the
order parameter satisfies any universal relation we plotted the EA
q values for different
concentrations vs. their respective scaled temperatures T/Tm. The data are shown in Figure
5.14. Interestingly we found that the
()
mEA TTq / curves for different concentrations (but
the same frequency) follow a universal pattern.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
qEA
T/Tm
BZT0.3
BZT0.4
BZT0.5
BZT0.6
qEA=1-(T/Tm)n
Figure 5.14: EA-order parameter EA
q as a function of scaled temperature T/Tm for various BZTx
compositions and f = 500 kHz. The degree of overlap of the curves below T = 0.85 Tm and the
deviations at higher T are evident. The solid curve indicates the scaling function
()
n
mEA TTq /1 −=
with n = 2.05 ± 0.1.
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 94
This behavior is evident from Figure 5.14, where the curves for the four different
concentrations overlap to a very high degree up to T~ 0.85Tm. Above this temperature
range all the curves show a clear tail with EA
q vanishing at T significantly higher than Tm.
The non-vanishing of EA
q at T = Tm and the deviation from the extrapolated fit at higher
temperatures is not surprising since the ordered regions or PNRs are understood to exist at
temperature well beyond the Curie point, up to the Burns temperature [9, 161]. However
their numbers are too few or the interactions between them are too weak to develop a
correlated behavior.
Typically spin glass systems have been shown to follow a temperature dependence
of universal sort given as q~
()
n
g
TT /1−, where the value of n can vary depending on the
dimensionality of the system, the assumed form of the random field distribution and the
proximity to the critical temperature [182, 183]. In our case we found that the
()
mEA TTq /
dependence that most closely describes the universal behavior as shown by the solid line
in Figure 5.14 and has the dependence
()()
n
mmEA TTTTq /1/ −= with n = 2.0 ± 0.1. While
such dependence is not typical for spin glass systems it has been observed in an
unconventional spin glass Mg1+tTitFe2-2tO4 [184]. At this point the scaling behavior that
we observe can only be justified as an empirical fit [185].
The observed behavior of EA
q has also been compared to the predictions of the
spherical random bond-random field (SRBRF) model which has been applied to
heterovalent relaxors [55]. Firstly this model predicts a linear T dependence for EA
q at low
temperatures, if one is to ignore random fields. We however do not find linear EA
q(T/Tm)
dependence for any appreciable temperature range. Secondly, if random fields are
considered as playing a significant role, then the SRBRF model predicts a systematic
change in the EA
q(T/Tm) behavior with increasing random field strength i.e. the absence of
a universal scaled behavior. We on the other hand find an overlapping or universal EA
q
(T/Tm) behavior for the various compositions. We understand the higher x compositions
as corresponding to higher level of random field strengths Δ [55]. Thus if random fields
were to be playing a significant role in determining the order in these relaxors then the
EA
q(T/Tm) curves for different x should not overlap. Therefore we emphasize that our data
show that in case of homovalent substituted relaxors, the role of random fields is not
substantial. This is also in agreement with the results of [129] where they found that the
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 95
relaxor behavior continues in BZT0.5 even if they completely switched off the random
fields in their simulations. They have argued that small antiferroelectric contributions
present in this system may initiate the observed relaxor behavior.
The analysis of the data up to this point has been in terms of the development of
correlations for a fixed frequency and as a function of temperature. We further addressed
the question of the dynamical behavior of the order parameter by comparing the EA
q
(T/Tm) behavior over a wide range of frequencies. The order parameter variation with
temperature was extracted for each frequency from the data of Figure 5.11 (a-d). The
behavior for each composition is shown separately in Figure 5.15 (a-d) and each curve in
the respective figures corresponds to a particular measuring frequency.
Figure 5.15: EA-order parameter EA
q for various compositions BZTx as function of reduced
temperature T/Tm measured at frequencies of 0.2-500 kHz.
0.0 0.5 1.0 1.5 2.0 2.5
(d)BZT0.6
T/T
m
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
(c)BZT0.5
q
EA
T/T
m
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
0.0
0.2
0.4
0.6
0.8
1.0
q
EA
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
(a) BZT0.3
(b) BZT0.4
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 96
It is apparent from Figure 5.15 (a) that EA
q(T/Tm, f) curves for a particular
composition show a frequency dependence in general, while for x= 0.3 the curves for
different frequencies just overlap. We note also that x= 0.3 is the composition where the
relaxor behavior is known to just set in for the BZT system. This may also be understood
as the concentration where polar cluster sizes grow large enough and correlations set in
between them. As x increases, we note that the scaled EA
q(T/Tm, f) demonstrates an
increasing level of frequency dependence, as evident from Figure 5.15 (a-d).
We see that the high frequency curves lie systematically below the low frequency
curves. In other words, for a fixed composition and fixed value of the scaled temperature,
smaller order parameter EA
q(T/Tm, f) values are associated with the higher frequencies.
This obviously suggests that the degree of correlation between the PNRs decreases with
increasing frequency. Note also that as we move to higher concentrations (Figure 5.15 (b,
c, d)) and thereby deeper into the relaxor state, the gap between the low frequency and the
high frequency curves increases. This can be interpreted in the sense that for larger x
compositions there is a broader distribution of relaxation times which is reflected in a
greater level of dispersion [72]. We also note that for a fixed composition x, the onset
point T*/Tm, where EA
q(T/Tm, f) initially assumes a non-zero value, shifts to lower scaled
temperatures with increasing frequencies. This indicates that with increasing frequency
less time is available for the correlations to develop resulting in low values of the order
parameter EA
q(T/Tm, f).
Figure 5.16 shows, for different compositions, the variation in the onset point of
the EA-order parameter as a function of frequency. It is clear from the figure that the
frequency dispersion increases with increase in the Zr content. The maximum change in
the onset point ⎥
⎦
⎤
⎢
⎣
⎡
Δ
m
T
T*and change in the value of 1/ =
Δm
TT
EA
qin the measured frequency
range are defined as
kHz
m
kHz
mm T
T
T
T
T
T
5002.0
*−=
⎥
⎦
⎤
⎢
⎣
⎡
Δ and kHz
EA
kHz
EA
TT
EA qqq m5002.01/ −=Δ = 5.9
Figure 5.17 shows the change in the onset point is increasing with increase in the Zr
concentration. Figure 5.17 also shows the change in the values of 1/ =
Δm
TT
EA
qextracted
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 97
from the data of Figure 5.15 for the measured frequency range. At a fixed temperature Tm
the maximum change 1/ =
Δm
TT
EA
qalso increases with increase in the Zr concentration.
4 6 8 10 12 14
1.0
1.2
1.4
1.6
1.8
2.0
2.2
T* / Tm
ln (f)
BZT0.3
BZT0.4
BZT0.5
BZT0.6
Figure 5.16: Frequency dependence of the onset temperature T*/Tm of EA-order parameter EA
q for
various compositions BZTx.
0.30 0.35 0.40 0.45 0.50 0.55 0.60
0.0
0.2
0.4
0.6
0.8
1.0
Δ
[T / Tm]
Δ
q |T/Tm =1
Zr. Conc. (x)
Δ
[T */ Tm]
0.1
0.2
0.3
0.4
0.5
0.6
Δ
q |T/Tm =1
Figure 5.17: Variation in change in onset point and change in EA
q measured at 0.2 kHz and 500 kHz
frequency as function of Zr content.
Chapter 5 RELAXOR FERROELECTRICS: ZIRCONIUM DOPED BARIUM TITANATE
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 98
5.3 Summary
The dielectric properties of bulk ceramic compositions with general formula
BaZrxTi1-xO3 (0.3 ≤ x ≤ 0.8) and perovskite structure were studied. The temperature
dependent dielectric studies were carried out on the compositions BaZrxTi1-xO3 (0.3 ≤ x ≤
0.8) in the temperature range 300 K to 10 K and frequencies 0.2 to 500 kHz.
Strong frequency dispersion is observed around the ɛʹm and ɛʺm peak for all the
BaZrxTi1-xO3 compositions. Similarly we find for all the BaZrxTi1-xO3 compositions that
with increasing frequency εʹm decreases and the peak temperature Tʹm is shifted to higher
temperatures. In the same manner the temperature of the loss peak εʺm increases with
increase in frequency whereas the peak value increases. The compositions show a diffuse
phase transition having the respective Curie temperatures much below the room
temperature. This is a typical characteristic of relaxor ferroelectrics.
A clear deviation from Curie-Weiss law at Tdev is observed for all the compositions.
To study the diffuseness, the data were fitted with a modified Curie-Weiss law and the
degree of diffuseness was also calculated. We have studied how the relaxor behavior
gradually evolves with the increasing substitution of Zr+4 ions for the Ti+4 in the matrix of
BaTiO3. Relaxor behavior becomes more prominent with the increase in content of Zr+4
ions i.e. with increasing amount of the polar nano regions.
The dielectric relaxations in BZTx ceramics are found to follow Vogel-Fulcher type
behavior and the experimental data were found to be in good agreement with the
proposed model.
In the last section, we have presented a detailed study of the BZTx relaxor system
with the view to relate the dielectric response of this system with the picture of a dipolar
glassy system. The behavior is seen to be well described by the development of a mean
field order parameter. For a fixed frequency a universal behavior of the order parameter is
reported for a range of relaxor concentrations. This universal behavior also supports the
argument that in these isovalent substituted relaxor systems the random fields may not
play a significant role. The dynamic behavior of the order parameter is also consistent
with a weakening of the correlations between the ordered regions with increasing
frequency.
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 99
Chapter 6 RELAXOR PROPERTIES OF
COMPOSITE CERAMICS
This chapter deals with the dielectric properties of the [BaZr0.5Ti0.5O3]0.65: [CoFe2O4]0.35
composite material. We report the effect of magnetic field on the dielectric response in
a relaxor ferroelectric and ferromagnetic composite [BaZr0.5Ti0.5O3]0.65: [CoFe2O4]0.35.
Relaxor characteristics such as dielectric peak temperature and activation energy show
a dependence on applied magnetic fields. This is explained in terms of increasing
magnetic field induced frustration of the polar nano regions comprising the relaxor.
The results are also consistent with the mean field formalism of dipolar glasses. It is
found that the variation of the spin glass order parameter qEA(T) is consistent with
increased frustration and earlier blocking of nano polar regions with increasing
magnetic field.
Recently two developments have taken place in the field of ferroelectrics which are
intriguing from the perspective of basic and applied physics. The first of these is the
explanation of relaxor or glassy features in the dielectric response in analogy with the
behavior of their magnetic counterparts spin glasses [130, 186]. Secondly the
development of composite materials that include multiferroic materials whereby one
component is ferroelectric and the other is ferromagnetic [187]. A number of studies have
been reported on the magnetic field dependence of ferroelectric multiferroics where the
dielectric response is controlled by applied magnetic fields, with manifest applications
[134, 188, 189]. It is understood that the magnetoelectric coupling in such composite
systems arises from the lattice strains developed by the magnetic component on the
application of a magnetic field [190]. However comparable studies on relaxor
multiferroics that highlight the effects of magnetic fields on the dielectric, and in
particular the relaxor features, have not been reported to date. This constitutes the main
focus of this chapter.
Considering the relaxor ferroelectric as a system with in-built randomness of
coupling between polar nanoregions, we can envisage a further disorder in the composite
due to the randomness of the couplings between the polar-nanoregions and the magnetic
component. In this chapter we address the question of how the applied magnetic fields
affect the relaxor properties of a typical relaxor-ferrimagnetic composite and go on to
show that these effects can be described within the framework of the mean field theories
of spin glasses applied to relaxor systems.
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 100
The base ferroelectric material selected for this study is the well-studied perovskite
Barium Titanate BaTiO3 (BTO) [4, 124, 191], which can form solid solution with
additives such as Zr, Sn or Sr [9, 192]. Relaxor properties are introduced on the addition
of sufficient concentration of Zr (substituting for Ti). For Zr concentrations greater than
25% typical relaxor behavior sets in [4, 124, 191]. The relaxor composition chosen was
BaZr0.5Ti0.5O3, which is a type II relaxor where the relaxor behavior is strain mediated
[193]. For the ferromagnetic component we have selected CoFe2O4 (CFO) due to its
excellent magnetostrictive properties. The composite ratio (BZTx : CFO) studied in this
work was 65:35.
6.1 Magnetic Measurements
We present here the magnetic characterization of CoFe2O4 and [BaZr0.5Ti0.5O3]0.65:
[CoFe2O4]0.35 bulk samples. The magnetic characterization of these bulk samples was
performed by vibrating sample magnetometer (VSM) at room temperature with maximum
applied field up to 7 kOe. The hysteresis M(H) loops for both the samples at room
temperature for an applied field of ±7 kOe are presented in Figure 6.1.
-8000 -6000 -4000 -2000 0 2000 4000 6000 8000
-80
-60
-40
-20
0
20
40
60
80
BZ65C35
M (emu/g)
H (Oe)
CFO
Figure 6.1: Magnetic hysteresis curves of samples measured at room temperature.
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 101
The M(H) loops depicts a typical ferromagnetic behavior in these samples. The
difference between the saturation magnetization, remanent magnetization and coercivity
values are clear from the figure. The values of MS and Mr and HC extracted from these
loops are listed in Table 6.1. The measured values of MS, Mr and HC for CFO are in good
agreement with those reported in the literature [194-196]. The MS values obtained for our
CFO sample is 77 emu/g (measured at 7 kOe) is slightly less than the reported values of
80 emu/g (measured at 20 kOe).
Table 6.1: Values of saturation magnetization MS measured at 7 kOe maximum filed, remanent
magnetization Mr and Coercivity HC of the CFO and BZ65C35 samples at room temperature.
Sample M
S
(emu/g) M
r
(emu/g) H
C
(Oe)
CFO 77 15 237
BZ65C35 20.5 4.4 400
The value of saturation magnetization in the composite sample is understandably
much less than that of the CFO sample. Since the CFO content in this sample is 35%, the
value of MS should be 35% of CFO values which is 27 emu/g. The observed value is even
smaller than this value. This degradation of the MS in case of composites may be
attributed to lack of correlation between the domains or due to the poor contacts due to
the formation of a magnetically dead layer between the grains. The coercivity of the
composite sample was found to be larger than that for the CFO sample, which may due to
the presence of more domain wall pinning defects in the samples [197].
6.2 Dielectric Spectroscopy
The temperature dependence dielectric spectrum (ε = εʹ+iε″) measured at 500 kHz,
and at an applied AC field of 50 mV for the [BaZr0.5Ti0.5O3]0.65: [CoFe2O4]0.35 sample is
shown in Figure 6.2. From the figure it is clear that the real component of the dielectric
permittivity εʹ increases gradually, passes through a broad maximum and then smoothly
decreases with increasing temperature. The broad maximum in εʹ is an important
character of the disordered structure with a diffuse phase transition (DPT). The imaginary
part ε″ also shows a broad peak at temperature lower than the peak temperature in the εʹ.
The non-equality of Tʹm and T
ʺm is well known for DPT and is observed by us in the
composite BZ65C35.
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 102
0 50 100 150 200 250 300
40
60
80
100
120
140
Temperature (K)
ε
'
0
2
4
6
8
10
12
ε
'
ε
"
ε
"
Figure 6.2: Dielectric spectrum of (BaZr0.5Ti0.5O3)0.65 : (CoFe2O4)0.35 measured at 500 kHz.
The maximum value of real part of dielectric susceptibility as determined from the
graph is εʹm = 131.6 while the peak temperature is Tʹm = 124 K. The maximum values for
the imaginary part and its respective peak temperature are εʺm = 9.51 and Tʺm = 78 K.
6.1.1 Frequency Dispersion in Dielectric Susceptibility
To highlight the effect of the CoFe2O4 in the composite system we compared the
dielectric spectrum of the composite sample with the pure relaxor ferroelectric sample.
We also performed the dielectric measurement on the composite system in the presence
of magnetic field, by placing the sample and cryostat in the DC magnetic (for detail see
chapter 3). The temperature dependence of the dielectric spectra of BZT0.5 (pure) and
BZ65C35 (composite) in the frequency range 0.2 to 500 kHz for zero applied fields are
shown in Figure 6.3 (a, b).
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 103
Figure 6.3: Temperature dependent dielectric spectra (εʹ, εʺ ) of the (a) BZT0.5, (b) BZ65C35 (H = 0
kOe), (c) BZ65C35 (H = 5 kOe) and (d) BZ65C35 (H = 7 kOe).
For both samples BZT0.5 and BZ65C35 the real component of the permittivity εʹ
passes through a broad maximum displaying the important characteristic of a relaxor
transition. Equally importantly, strong frequency dispersion is also evident in both
samples with a clear shift in the position of the maxima with frequency. With increasing
frequency the peak position shifts to higher temperature. Comparing the data for different
frequencies we note that the data are coincident down to about 150 K below which they
separate out. We also note that the maximum value of dielectric constant (εʹm) decreases
with increasing frequencies for both the compositions. Furthermore, the peak temperature
(Tʺm) of the imaginary part or loss component maxima (εʺm) was also frequency
dependent, increasing with increasing frequencies. These observations correspond to
those of a typical relaxor and agree with the reported trend in BZT0.5 [9]. In the case of
the composite sample Figure 6.3(b), it is evident that the overall value of the dielectric
constant has decreased as compared to the pure BZT0.5 sample and the peaks in εʹ and εʺ
have shifted to lower temperatures. The decrease of the dielectric constant and the peak
temperature in the case of composites may be attributed to a lack of correlation between
0 50 100 150 200 250 300
40
60
80
100
120
140
160
(d) BZ65C35
H = 7 kOe
Temperature (K)
ε
'
0
5
10
15
20
25
30
35
ε
"
0 50 100 150 200 250 300
40
60
80
100
120
140
160
(c) BZ65C35
H = 5 kOe
Temperature (K)
ε
'
0
5
10
15
20
25
30
35
ε
"
40
60
80
100
120
140
160
(b)
BZ65C35
H = 0 kOe
ε
'
0
5
10
15
20
25
30
35
ε
"
0
100
200
300
400
500
600
(a) BZT0.5
0.2 kHz
1 kHz
10 kHz
100 kHz
500 kHz
ε
'
0
50
100
150
200
250
ε
"
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 104
the polar nano regions or to poor contacts due to the formation of a dead layer between
the grains [198]. It is also possible that the strain induced by the cobalt ferrite part may
shift the dielectric peak to lower temperatures. It has also been shown that the position of
the dielectric maxima and other relaxor features may be affected by the sample
preparation conditions [199, 200]. However the relaxor behavior shown by the BZT0.5
sample is essentially preserved in the composite sample BZ65C35. Figure 6.3 (c, d) shows
the behavior of dielectric susceptibility for the BZ65C35 composite measured in the
presence of the DC magnetic field with values of 5 and 7kOe respectively. The effects of
the magnetic field on the dielectric susceptibility will be discussed in detail in section 6.3.
6.1.2 Dielectric Spectra of BaZrxTi1-xO3 and Curie Weiss Behavior
In normal ferroelectrics the high temperature dielectric behavior is described by the
Curie-Weiss Law given by Eq. 5.2 in temperature range T > TC. The fit of our data to Eq.
5.2 is displayed in Figure 6.4 (a-c) for the measuring frequency of 500 kHz. The
temperature Tdev where the data starts deviating from the Curie-Weiss law is marked in
Figure 6.4 (a-c). The values of the critical parameters C and θ of BZ65C35 sample along
with the values for BZT0.5 (from chapter 5) are given in Table 6.2. The negative value of
θ for the BZ65C35 sample is evident from the table. The negative values of θ may reflect
the antiferroelectric interactions present in the composite system or this may suggest
some secondary influential features (such as quantum fluctuations etc.), near and below
the relaxation peak [201]. However, it is seen that the dielectric permittivity of BZ65C35
sample follows the Curie–Weiss law at temperatures much higher than Tʹm. The
deviations from the Curie Weiss behavior are understood to be due to interactions among
the polar nanoregions that start well above the peak temperature. For both the y=0 and
y=0.35 compositions (pure relaxor and composite) the extent of deviation from the Curie-
Weiss behavior, as illustrated by the parameter ΔTm, where mdevm TTT −=Δ are shown in
Table 6.2. It is apparent that the presence of CoFe2O4 in the composite BZ65C35 sample
affects the values of peak temperatures (Tʹm and Tʺm), the deviation temperature Tdev and
the degree of deviation ΔTm in comparison to the values of pure BZT0.5 sample.
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 105
Figure 6.4: (a-c) 1/εʹ vs. temperature measured at 500 kHz. Straight line shows the Curie-Weiss behavior at
higher temperature. The lines are the fit to the Curie-Weiss law (equation 6.1). (d-f) Plot of ln(1/εʹ-1/εʹm)
vs. ln(T-Tʹm) of the data taken at 500 kHz applied frequency. The lines are the fit to the Modified
Curie-Weiss law (equation 6.3).
Table 6.2: The Critical parameter (Curie-Weiss constant C, Curie-Weiss temperature θ, the deviation
temperature Tdev, peak temperature Tm and the degree of deviation ΔTm) calculated from the Curie-
Weiss law fit of the experimental data in the high temperature paraelectric regime.
Sample C θ (K) Tde
v
(K) Tm (K) ΔTm (K)
BZT0.5 3.33×104 91 ± 2 234 146 88
BZ65C35|H=0 kOe 5.0×104 -232.5 ± 1 157 124 33
BZ65C35|H=5 kOe 5.0×104 -247.5 ± 2 157 128 29
BZ65C35|H=7 kOe 5.0×104 -254.2 ± 2 157 130 27
0123456
-14
-12
-10
-8
-6
-4
(f) H= 7 kOe
γ
= 1.63
Experimental data
MCW law fit to the data
ln(1/
ε
'-1/
ε
'
m
)
ln(T-T'
m
)
-14
-12
-10
-8
-6
-4
Experimental Data
MCW law fit to the Data
(e) H= 5 kOe
γ
= 1.63
ln(1/
ε
'-1/
ε
'
m
)
-14
-12
-10
-8
-6
-4
(d)H= 0 kOe
γ
= 1.60
ln(1/
ε
'-1/
ε
'
m
)
Experimental Data
MCW law fit to the Data
0 50 100 150 200 250 300
0.000
0.005
0.010
0.015
0.020
T
dev
(c) H= 7 kOe
Experimental Data
CW law fit to the Data
1/
ε
'
Temperature (K)
0.000
0.005
0.010
0.015
0.020
T
dev
(b) H= 5 kOe
Experimental Data
CW law fit to the Data
1/
ε
'
0.000
0.005
0.010
0.015
0.020
T
dev
(a) H= 0 kOe
Experimental Data
CW law fit to the Data
1/
ε
'
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 106
It is also known that a larger range of temperatures, from close to the peak and
extending to high temperatures, can be described in relaxors by the modified Curie-Weiss
relation (Eq. 1.10).
Figure 6.4(d-f) show a graph between ln[(1/ε)- (1/εm)] and ln(T-Tʹm) at 500 kHz. By
fitting the data using Eq. 5.4, we obtain values of γ for BZ65C35 and are listed in Table
6.3. These values lying in the range of 1.6-1.7 confirm typical relaxor behavior in both
the pure and composite samples and correlate well with the reported values in the
literature. For example A. Dixit et al. [202] have reported γ varying in the range 1.3-1.7
for thin films of BaZrxTi1-xO3 for (0.3 < x < 0.7). Similarly T. Maiti et al [201] reported
γ = 1.89 for ceramic of same composition. We also studied the behavior in the presence
of a magnetic field and the corresponding values of the parameters are also shown in
Table 6.3. It is seen that no appreciable change occurs in the values of these parameters as
a function of magnetic field.
Table 6.3: The critical parameters (Modified Curie-Weiss constant C1 and the diffuseness exponent γ)
calculated from the Modified Curie-Weiss law fit of the experimental data in the temperature range
above Tʹm.
Formula C1 -γ
BZT0.5 1.39×106 1.67
BZ65C35|H=0 kOe 1.29×106 1.60
BZ65C35|H=5 kOe 1.25×106 1.63
BZ65C35|H=7 kOe 1.10×106 1.63
6.1.3 Relaxation Time and Vogel-Fulcher Law
We now consider the behavior of the dielectric response below the peak
temperature. The observed decrease in the dielectric constant at lower temperatures is
typically associated with the freezing of the polar nano regions. These regions are
understood to begin forming at elevated temperatures well above the dielectric peak [9].
This freezing or long relaxation times are similar to spin glass freezing in magnetic
systems where the spin may not have enough thermal energy to overcome the competing
interactions. Thus the dynamics of the polar nanoregions slow down as the temperature is
lowered and can be represented by the Vogel-Fulcher relation (Eq. 1.11) [192].
The data for Tʹm (and Tʺm) as a function of frequency in the range 0.2-500 kHz are
shown in Figure 6.5. The fits of experimental data to Eq. 1.11 are also shown in the
Figure 6.5.
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 107
Figure 6.5: ln(f) vs. Tm curves of BZT0.5 and BZ50C35 samples. Lines represent the fitted curve
using Vogel Fulcher law (Eq. 1.11).
These fits suggest a Vogel-Fulcher (VF) dielectric relaxation in the samples. The
fitting parameters obtained are included in Table 6.4 and Table 6.5. The VF relation
followed by these samples essentially manifests the presence of short range interactions
between the polar nano regions, which in turn control the fluctuations of the polarization
above the freezing temperature, analogous to the case of spin glasses.
Table 6.4: The critical parameters, pre-exponential factor fₒ, the relaxation time τₒ, activation energy
Ea and static freezing temperature TVF for BZ65C35 obtained from Vogel-Fulcher law (Eq. 1.11) fit
for the peak temperature (Tʹm) variation with frequency. Data taken in presence of applied magnetic
fields (0, 5 and 7 kOe).
Formula fₒ (Hz) τₒ (sec) Ea (meV) TʹVF (K) ΔTʹrelax (K)
BZT0.5 8.40×109 1.19×10-10 94 33.6
50
BZ65C35|H=0 kOe 5.93×108 1.69×10-9 39 61.5
32
BZ65C35|H=5 kOe 5.66×109 1.77×10-10 63 51 34
BZ65C35|H=7 kOe 6.59×1010 1.52×10-11 84 48 32
50 60 70 80 90 100 110 120 130
(b)
BZ65C35
H = 0 kOe
T
m
(K)
60 80 100 120 140
4
6
8
10
12
14
(c)
BZ65C35
H = 5 kOe
ln (f)
T
m
(K)
60 80 100 120 140
(d) BZ65C35
H = 7 kOe
T
m
(K)
70 80 90 100 110 120 130 140 150
4
6
8
10
12
14
T'
m
T"
m
VF law Fit to Data
(a) BZT0.5
ln (f)
T
m
(K)
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 108
Table 6.5: The critical parameters, pre-exponential factor fₒ, the relaxation time τₒ, activation energy
Ea and static freezing temperature TVF for BZ65C35 obtained from Vogel-Fulcher law (Eq. 1.11) fit
for the peak temperature (Tʺm) variation with frequency. Data taken in presence of applied magnetic
fields (0, 5 and 7 kOe).
Formula fₒ (Hz) τₒ (sec) Ea (meV) TʺVF (K) ΔTʺrelax (K)
BZ50 8.40×109 9.55×10-12 101 15.8
38
BZ65C35|H=0 kOe 1.94×108 5.16×10-9 23.61 36.89 22
BZ65C35|H=5 kOe 5.99×1011 1.66×10-12 77.28 18.97 22
BZ65C35|H=7 kOe 2.15×1010 4.65×10-11 82.68 5.39 34
The examination of our data from Table 6.4 and Table 6.5 shows that the value of
TʹVF > TʺVF for all the samples. The values of the Debye frequencies fₒ or the relaxation
time τₒ are in good agreement with the previously reported data on these compositions.
The effect of the magnetic field on the activation energy and the time constant will
discuss in the next section.
6.3 Magnetic Field Effect on the Dielectric Susceptibility of
[BaZr0.5Ti0.5O3]0.65:[CoFe2O4]0.35
We now turn to the most important and novel feature of the current work, that is the
influence of magnetic fields on the dielectric properties of the composite sample.
0 50 100 150 200 250 300
40
60
80
100
120
140
H = 0 kOe
H = 5 kOe
H = 7 kOe
Temperature (K)
ε
'
0
10
20
30
40
ε
"
Figure 6.6: Dielectric spectrum (εʹ, εʺ) of BZ65C35 taken in the presence of 0 kOe, z 5 kOe and
c7 kOe magnetic field. Arrow indicates the direction of increasing field.
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 109
In these measurements the sample was cooled in a desired applied magnetic field (0,
5, or 7 kOe) and the capacitance of the sample was recorded as the temperature was
lowered. The dielectric constant for various applied magnetic fields and f = 500 kHz is
plotted in Figure 6.6.
The first prominent feature apparent in the figure is the decrease of the dielectric
constant with increasing field and this is evident over a wide temperature range. Note that
this effect of the magnetic field appears even at temperatures well above the peak
temperature and is nonvanishing, albeit slightly decreasing, at the maximum temperature
(300 K) studied. The second feature is that all the data for the different applied fields
merge at a certain temperature below the respective dielectric maxima. However the most
interesting observation evident here is a significant shift in the dielectric peak position to
higher temperatures with increasing magnetic field. To verify the relationship of this shift
with the applied field we show the effect of the field on the imaginary part of the
dielectric constant as well. As is evident these data also display a systematic shift of
dielectric loss peak with the application of magnetic field. Note also that we did not
observe any effect of magnetic field in the dielectric response of pure (non-magnetic)
BZT0.5 samples.
This observation testifies to the crucial role played by the magnetic component of
the composites in the above described effects. We also note that the differences between
the zero field and applied field data persist for temperatures above the peak. This is not
surprising considering the various reports [9] that confirm the existence of polar
nanoregions well above the relaxor peak temperature and considering that in our
temperature range (T<300 K) the ferrite is magnetically ordered. We expect that for
T>Tdev, where the PNRs are non-interacting, the individual nanoregions are expected to
experience the effects of the magnetic field induced strains. In contrast for T<Tdev the
interacting polar nano regions would experience the effects of the magnetic field induced
strains and this would be reflected in the field dependence of the dielectric constant for
Tm<T< Tdev, i.e. in the region where the PNRs are interacting but not yet locked into a
frozen state.
Similar experiments were done for other frequencies as well, covering the full
range from 0.2-500 kHz. These experiments demonstrate similar effects of applied
magnetic field on the dielectric constant. In Table 6.4 and Table 6.5 we have summarized
the effects of the frequency dependence of the peak temperatures in zero and applied
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 110
fields, along the lines of Vogel-Fulcher law analysis. We now discuss the results of those
fittings for the case of non-zero applied magnetic fields. From the values of the obtained
fit parameters shown in Table 6.4 and Table 6.5, we note that the values of the activation
energies increase very significantly with increased applied field. This suggests that the
activation of the dipolar orientations becomes more difficult as the applied magnetic field
increases. As the field is applied and increased, the resulting strains within the magnetic
component are transmitted to the coupled ferroelectric part, with randomness both in their
magnitude and directions. Thus in the presence of the applied magnetic field there is an
increased randomization of the interactions between the polar nano regions. The net effect
of these randomized strains, we understand, is to make a uniform orientation of the
various polar nano regions more difficult, leading to enhanced frustration and consequent
blocking of the dipoles at elevated temperatures. We can see from the data of Table 6.2
that the peak temperature Tm shifts consistently to slightly higher values with the
application of the magnetic field to the composite in line with the preceding discussion.
6.4 Magnetic Field Effect on the Order Parameter of
[BaZr0.5Ti0.5O3]0.65:[CoFe2O4]0.35
While the above discussion have been consistent with the general picture of a glassy
dipolar system analyzed along the lines of frustrated system, there is a need to put the
discussion on more firm ground. To this purpose we use the mean field description of
relaxor ferroelectrics and analyze the field dependence of the order parameter. In the
mean field description of relaxors [129, 130] the glass order parameter EA
q. In this
description EA
q can be taken to effectively represent the fraction of the total clusters
(polar nano regions) that are blocked and form a part of the glassy state. The values of
EA
q(T) were calculated using Eq. 5.8. The values of EA
q(T) thus obtained at various fields
are shown in Figure 6.7.
It is noticeable from these data that the value of EA
q for all the three cases starts
from zero at a temperature of about 162 K close to the temperature Tdev where the
respective εʹ(T) data begin to deviate from the pure Curie-Weiss behavior. The vanishing
of EA
q at T = Tdev has been reported in the literature [130]. We note also that on the low
temperature side the values of EA
q for all three cases merge in the temperature range 50-
60 K close to the TʹVF determined and described earlier (Table 6.4).
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 111
0 50 100 150 200
0.0
0.2
0.4
0.6
0.8
1.0
H = 0 KOe
H = 5 KOe
H = 7 KOe
qEA (T)
Temperature (K)
Tdev
Figure 6.7: Variation in Edward-Anderson order parameter EA
q at different applied magnetic field.
Arrow indicates the direction of increasing applied magnetic field.
The merging of all the curves (H ≥ 0) at low temperatures (~50 K) may be
associated with the freezing of the polar nano regions for T~TVF. Most significantly we
note that for TVF < T < Tm, as the field increases, the value of EA
q increases, for a fixed
temperature. To understand the significance of this observation we recall that the
magnitude of EA
q may be related to the fraction of the blocked clusters at a given
temperature. Hence the observation of an increasing value of EA
q for increasing H (for a
fixed T) indicates a higher fraction of blocked clusters. The field acts in this picture to
increase the randomness of the system by creating varying magnetostrictive strains that
affect various nanoregions in different and random ways. We therefore suggest that in
increasing magnetic fields the randomness of the magnetostrictive strain interactions
increases, hence increasing the frustration and thereby the fraction of blocked clusters.
The increase in the value of EA
q with magnetic field essentially reflects this dynamic.
Concluding, we have related our results to the description of relaxors using the
analogy with spin glasses. In this context our main observations viz. increase of the
Chapter 6 RELAXOR PROPERTIES OF COMPOSITE CERAMICS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 112
freezing temperature and the decrease of the dielectric constant with applied magnetic
fields find a consistent explanation. The field acts in this picture to increase the
randomness of the system by creating varying magnetostrictive strains that affect various
nanoregions in different and random ways. The consequent increased randomness of the
inter cluster interactions is expected to lead to a higher freezing temperature, in analogy
with the case of spin glasses where the freezing temperature is proportional to the
variance of the exchange constant distribution (Tf ~ΔJ). The increase of this order
parameter with increasing magnetic field also finds a consistent explanation within this
theory where the order parameter is a function of the scaled temperature T/ΔJ where ΔJ is
a parameter characterizing the exchange or dipolar interaction randomness. At a fixed
temperature and in increasing field (and randomness) the scaled temperature decreases,
and EA
q is expected to increase, as we see. Finally, our studies show that the ferroelectric
relaxor behavior may be controlled by external magnetic fields essentially due to the
transfer of strain from the magnetic phase to the relaxor phase and the effects can be
described within the mean field theory of dipolar glasses. This last result may be expected
to find interesting applications whereby unlike typical magneto-electric effects in
ferroelectrics, the magnetic field applied to the composite relaxors will affect the relaxor
attributes, as shown in this study.
Chapter 7 SUMMARY AND CONCLUSIONS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 113
Chapter 7 SUMMARY AND CONCLUSIONS
This chapter summarizes and concludes the findings of this work along with the future
directions.
The present work deals with the synthesis and electrical characterization of barium
zirconate titanate and their composites with cobalt ferrite. Detailed discussion and
conclusions have been given at the end of each chapter. Therefore, the concluding chapter
is devoted to the summarization of the main contributions of the work and arriving at
general conclusions.
The ceramic compositions with general formula BaZrxTi1-xO3 (with 0.3 ≤ x ≤ 0.8)
were synthesized by conventional sintering process. The room temperature X-ray
diffraction spectra of (BaZrxTi1-xO3) samples were found to be single phase and
correspond to a perovskite structure. The temperature dependent dielectric study on the
composition BaZrxTi1-xO3 (with 0.3 ≤ x ≤ 0.8) was carried out in the temperature range 10
to 300 K. Strong frequency dispersion is observed around the peak in the dielectric
susceptibility for all the BaZrxTi1-xO3 compositions. In all compositions we observe
decrease in the maximum value of ɛʹ with increasing frequency while the peak
temperature Tʹm is shifted to higher temperatures. In the same manner the peak
temperature of the loss maxima Tʺm increase with increase in frequency whereas the peak
value ɛʺm also increases, unlike the trend in ɛʹm. The compositions show a diffuse phase
transition having its Curie range of temperature much below the room temperature. This
is a typical characteristic of relaxor ferroelectrics. A clear deviation from Curie-Weiss
law at characteristic temperature Tdev is observed for all the compositions. To study the
diffuseness, the data were fitted with a modified Curie-Weiss law and the degree of
diffuseness represented by the parameter γ was also calculated. The systematic increase
of γ showed that the relaxor behavior gradually evolves with the increasing substitution of
Zr+4 ions for the Ti+4 in the matrix of BaTiO3. The dielectric relaxations in BZT ceramics
were found to follow Vogel-Fulcher type behavior (originally derived for the spin-glass
systems) and the experimental data were found to be in good agreement with the
theoretical fitting.
It is understood that the PNRs begin to be formed well above the peak temperature
and with decreasing temperatures; these independent regions begin to interact amongst
Chapter 7 SUMMARY AND CONCLUSIONS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 114
themselves. This trend was explored by calculating the Edward Anderson order parameter
()
TqEA ,
ω
from the experimental data of the dielectric measurement for BaTi1-xZrxO3 (0.3
≤ x ≤ 0.6).
The temperature and frequency dependence of the order parameter enabled us to
develop a picture of this system in parallel with that of frustrated spin glass like systems.
In particular the behavior was seen to be well described by the development of an order
parameter via mean field theory. For fixed frequency a universal behavior of the order
parameter was observed for a range of relaxor concentrations. This universal behavior
also supported the argument that in these isovalent substituted relaxor systems the random
fields may not play a significant role. The dynamic behavior of the order parameter i.e. a
smaller value at higher frequencies for the same scaled temperature was also consistent
with a weakening of the correlations between the ordered regions with increasing
frequency.
Furthermore we presented a novel method to control the relaxor features in these
systems by synthesizing a composite of magnetic and ferroelectric relaxor compositions.
Magnetoelectric relaxor composites of Ba(Zr,Ti)O3 with ferrites like CoFe2O4, were
prepared using conventional sintering techniques and were characterized. The peak of the
dielectric spectrum was found to shift to the low temperature in the composite samples as
compared to the pure relaxor sample, presumably as a result of the stress on the relaxor
components due to magnetostrictive effect of the ferrite component.
In this context our main observations were that we find an increase of the freezing
temperature and the decrease of the dielectric constant with applied magnetic fields. That
was a unique result in the sense of that it demonstrated the control of the dielectric relaxor
features through a magnetic field. We related our results to the description of relaxors
using the analogy with spin glasses. The magnetic field acts in this picture to increase the
randomness of the system, by creating varying magnetostrictive strains that affect various
nanoregions in different and random ways. The consequent increased randomness of the
inter cluster interactions is expected to lead to a higher freezing temperature, in analogy
with the case of spin glasses where the freezing temperature is proportional to the
variance of the exchange constant distribution, (Tf ~ΔJ). The increase of this order
parameter with increasing magnetic field finds a consistent explanation within this theory
where the order parameter is a function of the scaled temperature T/ΔJ where ΔJ is a
parameter characterizing the exchange or dipolar interaction randomness. At a fixed
Chapter 7 SUMMARY AND CONCLUSIONS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 115
temperature and in increasing field (and randomness) the scaled temperature decreased,
and EA
q was seen to increase. Another way of looking at the same trend was to note that
the magnitude of EA
q represented the fraction of blocked PNRs. Increasing field and
randomness of interactions between the PNRs was expected to lead to larger fraction of
these becoming blocked, at the same temperature. At the macroscopic level these studies
showed that the ferroelectric relaxor behavior may be controlled by external magnetic
fields essentially due to the transfer of strain from the magnetic phase to the relaxor
phase. We have shown that both the effects of the magnetic field in the composite
relaxor-ferrimagnetic system can be described within the mean field theory of dipolar
glasses.
Future Directions
To proceed further, an in depth analysis of the structure and microstructure of the
compound is essential. We plan to study the behavior of polar nano regions using TEM
for various low temperatures. This will help us to understand the growth and dynamic of
the polar nano region in the Ba(Zr,Ti)O3, furthermore the study of HRTEM may also shed
light on the interaction between these polar nanoregions. In addition to this the
temperature dependent Raman spectroscopy may also extract useful information to
understand the relaxor behavior. Unfortunately these tools are not available to us and we
may seek new collaborations in order to fulfil our goals. In this work we have also
conjectured that in case of isovalently substituted relaxor systems random fields may not
play a significant role. However for more clearer picture further work in this direction is
certainly required. For example to look in details the role of any antiferroelectric
interactions similar to work done by Akbarzadeh et. al. [129].
From application point of view, piezoelectric and pyroelectric measurements should
be carried out. Thin films can be prepared and studied for better understanding of the
relaxor behavior. We have initiated some work (not shown in this thesis), in the
fabrication of thin films in two different direction. Firstly we are growing thin films of
BZTx and CFO and plan to see how the magnetic coupling enhances either with number
of layers or layer thickness.
Further work can be carried out in the growth of the relaxor
ferroelectric/ferromagnetic self-assembled nanostructures thin films and their properties.
For example the growth of self-assembled nano rods in the BZTx matrix may be
Chapter 7 SUMMARY AND CONCLUSIONS
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 116
synthesized by phase segregation techniques used by various authors. For the formation
of Ba(Ti,Zr)O3-CoFe2O4 nanostructures or other systems, there are several interesting
questions to address: How to form highly ordered Ba(Ti,Zr)O3-CoFe2O4 nanostructures?
Ordered nanostructures are tremendously valuable for device fabrications. However, it is
still a challenge to grow well ordered complex oxides by self-assembly. The starting point
could be the deposition of the films on suitable substrates.
Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 117
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Studies of Ferroelectric and Multiferroic Behavior in [Ba(Zr,Ti)O3]1-y:[CoFe2O4]y System | 124
Apendex I. Published Papers Included in this Thesis
Paper 1. Magnetic Control of Relaxor Features in BaZr0.5Ti0.5O3 and CoFe2O4
Composite.
Muhammad Usman, Arif Mumtaz, Sobia Raoof and S. K. Hasanain.
Applied Physics Letters 102, 112911 (2013)
Paper 2. Order Parameter and Scaling Behavior in BaZrxTi1-xO3 (0.3 < x < 0.6)
Relaxor Ferroelectrics.
Muhammad Usman, Arif Mumtaz, Sobia Raoof and S. K. Hasanain.
Applied Physics Letters 103, 262905 (2013)
Paper 3. Response to the Comment on “Order Parameter and Scaling Behavior in
BaZrxTi1-xO3 (0.3 < x < 0.6) Relaxor Ferroelectrics”.
Muhammad Usman, Arif Mumtaz, Sobia Raoof and S. K. Hasanain.
Applied Physics Letters 104, 156103 (2014)
Note: The comment on the paper is also added for better understanding.
Comment on “Order parameter and scaling behavior in BaZrxTi1– xO3 (0.3 < x < 0.6)
relaxor ferroelectrics” [Appl. Phys. Lett. 103, 262905 (2013)]
J. Macutkevic1 and J. Banys
Applied Physics Letters 104, 156102 (2014)
Magnetic control of relaxor features in BaZr
0.5
Ti
0.5
O
3
and CoFe
2
O
4
composite
Muhammad Usman, Arif Mumtaz,
a)
Sobia Raoof, and S. K. Hasanain
Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan
(Received 19 January 2013; accepted 5 March 2013; published online 21 March 2013)
We report the effect of magnetic field on the dielectric response in a relaxor ferroelectric and
ferromagnetic composite (BaZr
0.5
Ti
0.5
O
3
)
0.65
-(CoFe
2
O
4
)
0.35
. Relaxor characteristics such as
dielectric peak temperature and activation energy show a dependence on applied magnetic fields.
This is explained in terms of increasing magnetic field induced frustration of the polar nanoregions
comprising the relaxor. The results are also consistent with the mean field formalism of dipolar
glasses. It is found that the variation of the spin glass order parameter q(T) is consistent
with increased frustration and earlier blocking of nanopolar regions with increasing magnetic field.
V
C2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4795726]
Ferroelectric materials have been known for quite a long
time and are acquiring increasing importance in modern
electronic industry where their unique properties are being
utilized in capacitors, piezoelectric transducers, and many
other interesting applications.
1
However, more recently two
developments have taken place in this field, which are in-
triguing from the perspective of basic and applied physics.
The first of these is the development of relaxor ferroelectrics
that exhibit glassy features in their dielectric response,
similar in many ways to their magnetic counterparts, spin
glasses.
2,3
Second, the development of composite materials
that include multiferroic materials whereby one component
is ferroelectric and the other is ferromagnetic.
4
A number of
studies have been reported on the magnetic field dependence
of ferroelectric multiferroics where the dielectric response is
controlled by applied magnetic fields, with manifest applica-
tions.
5
It is understood that the magnetoelectric coupling in
such composite systems arises from the lattice strains devel-
oped by the magnetic component on the application of a
magnetic field.
6
However, comparable studies on relaxor
multiferroics that highlight the effects of magnetic fields on
the dielectric, and in particular the relaxor features, have not
been reported to date. Typical characteristics of relaxor fer-
roelectrics are attributed to the formation of polar nanore-
gions within which the ferroelectric order is uniform,
7
while
charge, size, or structural disorder effectively leads to a
breakdown of the otherwise long range ordered regions into
these ordered and interacting/noninteracting nano-regions.
Considering the relaxor ferroelectric as a system with
in-built randomness of coupling between polar nanoregions,
we can envisage a further disorder in the composite due to
the randomness of the couplings between the polar-
nanoregions and the magnetic component. In the current
work, we address the question of how the applied magnetic
fields affect the relaxor properties of a typical relaxor-
ferrimagnetic composite and go on to show that these effects
can be described within the framework of the mean field the-
ories of spin glasses applied to relaxor systems.
The base ferroelectric material selected for this study is
the well-studied perovskite Barium Titanate BaTiO
3
(BTO),
8–10
which can form solid solution with additives
such as Zr, Sn, or Sr.
7,11
Relaxor properties are introduced
on the addition of sufficient concentration of Zr (substituting
for Ti), this results in the successive phase transitions of BTO
(cubic to tetragonal, tetragonal to orthorhombic, and finally
orthorhombic to rhombohedral) being pinched to a single, dif-
fuse phase transition.
12,13
For Zr concentrations greater than
25%, typical relaxor behavior sets in.
8–10
This is manifested
by a broad and frequency dependent peak in the dielectric
constant, as a function of temperature. The peak temperature
T
m
exhibits frequency dependence shifting to higher tempera-
tures with increasing frequencies. The relaxor composition
chosen was BaZr
0.5
Ti
0.5
O
3
, which is a type II relaxor where
the relaxor behavior is strain mediated.
14
As for the ferromag-
netic component, we have selected CoFe
2
O
4
(CFO) due to
its excellent magnetostrictive properties. The composite
ratio (BZT:CFO) studied in this work was 65:35. The
Ba(Ti
0.5
Zr
0.5
)O
3
(BZ50) and composite ceramic samples
(BaZr
0.5
Ti
0.5
O
3
)
0.65
-(CoFe
2
O
4
)
0.35
(BZ65C35) were prepared
by conventional sintering process described elsewhere.
15
The temperature dependence of the dielectric spectra of
BZ50 (pure) and BZ65C35 (composite) in the frequency
range 0.2 to 500kHz is shown in Fig. 1. For both samples
BZ50 and BZ65C35, the real component of the permittivity e0
passes through a broad maximum displaying the important
characteristic of a relaxor transition. Equally importantly,
strong frequency dispersion is also evident in both samples
with a clear shift in the position of the maxima with fre-
quency. With increasing frequency the peak position shifts to
higher temperature. Comparing the data for different frequen-
cies, we note that the data are coincident down to about 150K
below which they separate out. We also note that the maxi-
mum value of dielectric constant (e0
m) decreases with increas-
ing frequencies for both the compositions. Furthermore, the
peak temperature of the imaginary or loss component (e00
m)
was also frequency dependent, increasing with increasing fre-
quencies (not shown). These observations correspond to a
typical relaxor and agree with the reported trend in BZ50.
7
In
thecaseofthecompositesampleFig.1(b), it is evident that
the overall value of the dielectric constant has decreased as
a)
Author to whom correspondence should be addressed. Electronic mail:
arif@qau.edu.pk
0003-6951/2013/102(11)/112911/5/$30.00 V
C2013 American Institute of Physics102, 112911-1
APPLIED PHYSICS LETTERS 102, 112911 (2013)
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compared to the pure BZ50 sample and the peaks in e0have
shifted to lower temperatures. The decrease of the dielectric
constant and the peak temperature in the case of composites
may be attributed to a lack of correlation between the nanopo-
lar regions or to poor contacts due to the formation of a dead
layer between the grains.
16
It has also been shown that the
position of the dielectric maxima and other relaxor features
may be affected by the sample preparation conditions.
17
However, the relaxor behavior shown by the BZ50 sample is
essentially preserved in the composite sample BZ65C35.
In normal ferroelectrics, the high temperature dielectric
behavior is described by the Curie-Weiss law (T>T
CW
),
1
e0¼TTCW
C;(1)
where Cis the Curie-Weiss constant and T
CW
is the Curie-
Weiss temperature. The fit of our data to the above expres-
sion is displayed in Fig. 2for the measuring frequency of
500 kHz. The temperature T
dev
where the data starts deviat-
ing from the Curie-Weiss law is marked in Figs. 2(a) and
2(b).T
dev
was about 210 K and 188 K for the BZ50 and
BZ65C35 samples, respectively.
However, a larger range of temperature from close to
the peak and extending to high temperatures can be
described in relaxors by the modified Curie-Weiss relation,
1
e01
e0
m
¼ðTT0
mÞc
C1
;1c2:(2)
Here e0is the value of the permeability at the temperature T
while e0
mis the value at the peak temperature while cis the
parameter that relates to the character of the phase transition.
For the normal ferroelectrics, c¼1 and corresponds to the
normal Curie Weiss Law while for relaxors it is reported to
lie between 1 and 2, with c¼2 describing the completely
diffuse phase transition (DPT). Finally C
1
in Eq. (2) is the
modified Curie-Weiss constant.
The insets of Fig. 2show a graph between ln[(1/e0)-(1/
e0
m)] and ln(T-T0
m) at 500 kHz. By fitting the data using Eq.
(2), we obtain values of cto be 1.67 and 1.60 for BZ50 and
BZ50C35, respectively. These values confirm typical relaxor
behavior in the samples and correlate well with the reported
values in the literature. For example, Dixit et al.
18
have
reported cvarying in the range 1.3–1.7 for thin films of
BaZr
x
Ti
1x
O
3
for (0.3 <x<0.7). Similarly, Maiti et al.
19
reported c¼1.89 for ceramic of same composition.
Comparing the values of different parameters, e.g. c,T0
m, and
E
a
(see below) with those reported,
7,18,19
we presume that
our samples may be placed in the moderate relaxor category.
Considering the very close gamma values of the two samples
in this light strongly suggests that they can be regarded as
being at the same level of disorder with regard to the nature
of the phase transition.
We now consider the behavior of the dielectric response
below the peak temperature. The observed decrease in the
dielectric constant at lower temperatures is typically associ-
ated with the freezing of the polar nano regions. These
regions are understood to begin forming at elevated tempera-
tures well above the dielectric peak.
7
This freezing or long
FIG. 1. Temperature dependent dielectric spectra e0of the (a) BZ50 (b)
BZ65C35. Arrow indicates the direction of increasing frequency. 䊏0.2, 䊉
1, 䉱10, 䉲100, 䉬500 kHz.
FIG. 2. 1=e0vs. temperature measured at 500 kHz. Straight line shows the
Curie-Weiss behavior at higher temperature. Inset shows modified Curie-
Weiss law fitted for the respective samples.
112911-2 Usman et al. Appl. Phys. Lett. 102, 112911 (2013)
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relaxation times are similar to spin glass freezing in mag-
netic systems where the spin may not have enough thermal
energy to overcome the competing interactions. Thus the dy-
namics of the polar nanoregions slow down as the tempera-
ture is lowered and can be represented by the Vogel-Fulcher
relation,
11
f¼foexp Ea
kBðT0
mTVFÞ
:(3)
Here T
VF
is the static (f¼0) freezing temperature, E
a
is the
activation energy for polarization fluctuation of an isolated
nanopolar region, f
o
is a characteristic attempt frequency,
while T0
mis the peak temperature corresponding to the fre-
quency f. The data for T0
mas a function of frequency in the
range 0.2–500 kHz are shown in Fig. 3. Also shown are the
fits to Eq. (3). These fits suggest a Vogel-Fulcher (VF)
dielectric relaxation in the samples. The fitting parameters
obtained are included in Table Iand are comparable with
reported ones.
7,19
The VF relation followed by these samples
essentially manifests the short range interactions between the
polar nanoregions, which control the fluctuations of the
polarization above the freezing temperature, analogous to
the case of spin glasses.
We now turn to the most important feature of the current
work which is the influence of magnetic fields on the dielec-
tric properties of the composite sample. In these measure-
ments, the sample was cooled in a desired applied magnetic
field (0, 5, or 7 kOe) and the capacitance of the sample was
recorded as the temperature was lowered. The dielectric con-
stant for various applied magnetic fields and f¼500 kHz is
plotted in Fig. 4.
The first prominent feature apparent in the figure is the
decrease of the dielectric constant with increasing field and
this is evident over a wide temperature range. Note that this
effect of the magnetic field appears even at temperatures
well above the peak temperature and is nonvanishing, albeit
slightly decreasing, at the maximum temperature (300 K)
studied. The second feature is that all the data for the differ-
ent applied fields merge at a certain temperature below the
respective dielectric maxima. However, the most interesting
observation evident here is a significant shift in the dielectric
peak position to higher temperatures with increasing mag-
netic field. To verify the relationship of this shift with the
applied field, we show the effect of the field on the imaginary
part of the dielectric constant as well. As is evident these
data also display a systematic shift of dielectric loss peak
with the application of magnetic field. Note also that we did
not observe any effect of magnetic field in the dielectric
response of pure (non-magnetic) BZ50 samples. This obser-
vation testifies to the crucial role played by the magnetic
component of the composites in the above described effects.
We also note that the differences between the zero field and
applied field data persist for temperatures above the peak.
This is not surprising considering the various reports
7
that
confirm the existence of polar nanoregions well above the
relaxor peak temperature and considering that in our temper-
ature range (T<300 K) the ferrite is magnetically ordered.
We expect that for T>T
dev
, the onset temperature for devia-
tion from the Curie-Weiss behavior, the individual nanore-
gions are expected to experience the effects of the magnetic
field induced strains. For T<T
dev
the interacting nanopolar
regions would experience the effects of the magnetic field
induced strains and this would be reflected in the field de-
pendence of the dielectric constant for T>T0
m.
Similar experiments were done for other frequencies as
well, covering the full range from 0.2–500 kHz. These
experiments demonstrate similar effects of applied magnetic
field on the dielectric constant. While detailed analysis of the
field dependence in the high temperature region will be
reported elsewhere, here we report the results of fitting the
Vogel-Fulcher law to the variation of the peak temperatures
with frequency, with non-zero applied magnetic field. The
values of the obtained fit parameters are shown in Table I.
We note that the values of the activation energies increase
very significantly with applied field suggesting that the
FIG. 3. lnf vs. T0
mcurves of 䊏BZ50 and 䊉BZ50C35 samples. Lines repre-
sent the fitted curve using Vogel Fulcher law.
TABLE I. Parameters calculated from Vogel Fulcher fitting for BZ50 and
BZ65C35.
Parameters f
o
(Hz) E
a
(meV) T
VF
(K)
BZ50 8.40 10
9
94 33.6
BZ50C35j
H¼0kOe
5.93 10
8
39 61.5
BZ50C35j
H¼5kOe
5.66 10
9
63 51
BZ50C35j
H¼7kOe
6.59 10
10
84 48
FIG. 4. Dielectric spectra ðe0;e00 Þof BZ65C35 taken in the presence of 䊏
0 kOe, 䊉5 kOe, and 䉱7 kOe magnetic field.
112911-3 Usman et al. Appl. Phys. Lett. 102, 112911 (2013)
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activation of the dipolar orientations becomes more difficult
as the applied magnetic field increases. This observation sug-
gests that in the presence of the applied magnetic field there
is an increased randomization of the interactions between the
nanopolar regions. As the field is applied and increased, the
resulting strains within the magnetic component are trans-
mitted to the coupled ferroelectric part, with randomness
both in their magnitude and directions. The net effect of
these randomized strains, we understand, is to make a uni-
form orientation of the various nanopolar regions more diffi-
cult, leading to enhanced frustration and consequent
blocking of the dipoles at elevated temperatures. We quan-
tify these speculative arguments by referring to the mean
field description of relaxors
2,20
where the glass order param-
eter qis defined as qhPiPji, where Piand Pjare the
neighboring nanopolar regions dipole moments. In this
description, qcan be taken to effectively represent the frac-
tion of the total clusters (nanopolar regions) that are blocked
and form a part of the glassy state. Applying the Sherrington
and Kirkpatrick (SK) infinite rage model for the spin glass to
the dipolar glasses, the order parameter qcan be calculated
from
2,21
v¼Cð1qÞ
TTCW ð1qÞ;(4)
where Cand T
cw
are Curie Weiss constant and temperature
obtained from high temperature fitting data of the dielectric
constant in zero and applied magnetic fields, respectively. The
values of q(T) thus obtained at various fields are shown in Fig. 5.
It is noticeable from these data that the value of qfor all
the three cases starts from zero at a temperature of about
162 K close to the temperature T
dev
where the respective
e0(T) data begin to deviate from the pure Curie-Weiss behav-
ior. The vanishing of qat T¼T
dev
has been reported in the
literature.
2
We note also that on the low temperature side the
values of qfor all three cases merge in the temperature range
50–60 K close to the T
VF
determined and described earlier
(Table I). The merging of all the curves (H 0) at low tem-
peratures (50 K) may be associated with the freezing of the
nanopolar regions for TT
VF
. Most significantly we note
that, for T
VF
<T<T0
m, as the field increases, the value of q
increases for a fixed temperature. This suggests that our data
is consistent with the mean field description whereby in
increasing magnetic fields the randomness of the interactions
increases, thereby increasing the value of qof the composite
and implying that a greater fraction of the nanopolar regions
are blocked due to the application of the field.
Concluding, we have related our results to the descrip-
tion of relaxors using the analogy with spin glasses. In this
context, our main observations, viz. increase of the freezing
temperature and the decrease of the dielectric constant with
applied magnetic fields find a consistent explanation. The
field acts in this picture to increase the randomness of the
system by creating varying magnetostrictive strains that
affect various nanoregions in different and random ways.
The consequent increased randomness of the intercluster
interactions is expected to lead to a higher freezing tempera-
ture, in analogy with the case of spin glasses where the freez-
ing temperature is proportional to the variance of the
exchange constant distribution (T
f
DJ). The increase of this
order parameter with increasing magnetic field finds a con-
sistent explanation within this theory where the order param-
eter is a function of the scaled temperature T/DJ, where DJis
a parameter characterizing the exchange or dipolar interac-
tion randomness. At a fixed temperature and in increasing
field (and randomness) the scaled temperature decreases, and
qis seen to increase. Finally, our studies show that the ferro-
electric relaxor behavior may be controlled by external mag-
netic fields essentially due to the transfer of strain from the
magnetic phase to the relaxor phase and the effects can be
described within the mean field theory of dipolar glasses.
The authors would like to thanks the Higher Education
Commission of Pakistan for the financial support under the
project Development and Study of Magnetic Nanostructures.
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(䊏0 kOe, 䊉5 kOe and 䉱7 kOe).
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Order parameter and scaling behavior in BaZr
x
Ti
12x
O
3
(0.3 <x<0.6) relaxor
ferroelectrics
Muhammad Usman, Arif Mumtaz,
a)
Sobia Raoof, and S. K. Hasanain
Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan
(Received 8 November 2013; accepted 15 December 2013; published online 31 December 2013)
We report the relaxor behavior of the zirconium doped barium titanate BaZr
x
Ti
1x
O
3
solid solutions
and discuss the temperature, frequency, and concentration dependence in terms of correlations
among the polar nanoregions. The relaxor behavior is analyzed within the mean field theory by
estimating the Edward-Anderson order parameter qEA. Additionally, we find that qEA calculated for
the different concentrations obeys a scaling behavior qEA ¼1T=Tm
ðÞ
n,whereT
m
are the
respective dielectric maxima temperatures and n¼2.0 60.1. The frequency dependence of the qEA
also shows results consistent with the above mentioned picture. V
C2013 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4860967]
Ferroelectric relaxor systems and the understanding of
their complex behavior have recently attracted a lot of inter-
est. In part, this is due to their tunable dielectric properties
and potential applications in microwave devices, ferroelec-
tric random access memories (FERAMs), and multi layered
capacitors, to name a few.
1
Apart from technological appli-
cations there is also a strong interest in these materials as
typifying a complex form of polar order in a system with
manifest randomness.
2,3
Relaxor ferroelectrics are character-
ized, in particular, by a diffuse and frequency dependent
dielectric peak temperature. Perovskites, represented as
ABO
3
, are common ferroelectric materials and have been
extensively studied. In these perovskite structures relaxor
behavior sets in for certain substitutions at either A or B or
both A and B sites. Typical examples are PbMg
1/3
Nb
2/3
O
3
(PMN),
4,5
PbZn
1/3
Nb
2/3
O
3
(PZN),
5
and the lead (Pb) free
systems BaZr
x
Ti
1x
O
3
(BZTx) and BaTi
1x
Sn
x
O
3
(BSnT).
6
Generally speaking relaxor ferroelectrics may be divided
into two broad categories, pertaining to heterovalent or
homovalent substitutions. The systems with heterovalent
compositions such as PMN and their solid solutions, e.g.,
PbMg
1/3
Nb
2/3
O
3
-PbTiO
3
(PMN-PT)
7
show a well-defined
dielectric peak that exhibits relaxor features. The relaxor
behavior in these heterovalent systems is understood as
occurring mainly due to two reasons: the charge imbalance
arising due to the differing dopant valence and second from
the presence of ordered polar regions of nanosize.
8,9
Both
these effects produce random quenched fields, an essential
ingredient in canonical relaxor materials. A lot of work has
been done in such systems, and they are relatively well
understood. On the other hand, the homovalent substitution
of Ti
þ4
, for example, in BaTiO
3
, with Zr
þ4
does not produce
a significant charge imbalance and therefore is not expected
to produce quenched random fields.
2
In contrast to the heter-
ovalent relaxors the relaxor behavior in these latter materials
appears only after a significantly higher level substitution.
For example, in the case of BaZr
x
Ti
1x
O
3
the relaxor behav-
ior starts for x>0.27.
10
The origin of relaxor behavior in
such homovalent titanate material has recently attracted
attention for both experimental and theoretical studies.
It has been argued in the past and also in recent works
that ferroelectric relaxors have many features that resemble
those of magnetic spin glass systems and are describable gen-
erally by the models developed for those systems.
8
Akbarzadeh et al. in their numerical studies have described
the behavior of the dielectric susceptibility for the
BaZr
0.5
Ti
0.5
O
3
,
11
using the Edward-Anderson order parame-
ter.
12
Their work also illustrates the presence of polarization
clusters that grow in size and number with decreasing temper-
atures and begin to interact below a certain characteristic
temperature. These polarization clusters are identical to the
polar nanoregions (PNRs) that are understood to be central to
the relaxor behavior, with the addition that they are now con-
sidered to be dynamic entities, i.e., the center of the cluster
may shift from one Ti ion to another with time. The predic-
tions of this model were found to be in good agreement with
the experiments of Bhalla et al.
10
for the temperature and fre-
quency dependence of the dielectric constant of BZT materi-
als. This is consistent with the generally accepted picture of
the development of polar nanoregions, their growth, and the
development of interactions between them. However, it
remains to be seen how (and if) these features manifest them-
selves experimentally in the order parameter, both with
increasing levels of substitution and increasing frequency. In
the current work we have studied the BaZr
x
Ti
1x
O
3
relaxor
ferroelectrics (0.3 x0.6) and present our results for the
dielectric susceptibility as a function of concentration, tem-
perature, and frequency (x,T,f) and analyze the variations of
the obtained Edward-Anderson (EA) order parameter for
these variables (x,T,f). We further explore the important
question whether the order parameter for different concentra-
tions can be described by a universal behavior, qEAðT=TmÞ,as
is typically seen in classical spin glasses. In a similar vein the
variation of qEA(T,f) is studied to determine how the dynami-
cal response at different temperatures manifests itself in the
order parameter.
The BaZr
x
Ti
1x
O
3
bulk ceramic samples with different
Zr concentration (0.3 x0.6) were prepared by conven-
tional solid state reaction with a two-step sintering process,
a)
Author to whom correspondence should be addressed. Electronic mail:
arif@qau.edu.pk
0003-6951/2013/103(26)/262905/5/$30.00 V
C2013 AIP Publishing LLC103, 262905-1
APPLIED PHYSICS LETTERS 103, 262905 (2013)
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On: Tue, 31 Dec 2013 15:27:52
and the preparation and characterization of the obtained sin-
gle phase structures have been detailed elsewhere.
13
Dielectric measurements were carried out over a frequency
range 0.2–500 kHz using Wayne Kerr LCR meter
(WK-4275) in the temperature range of 10–300K using a he-
lium closed cycle system (Janis CCS-350 Cryostat).
In Fig. 1, both real and imaginary parts of the dielectric
susceptibility measured at various frequencies as a function
of temperature are shown for different Zr concentrations. For
all concentrations studied, clear frequency dispersion can be
seen. It is also evident from the figure that the dielectric peak
shifts to low temperature as the Zr concentration increases.
Furthermore peak broadening with increasing Zr concentra-
tion is also evident. These observations are consistent with
many reported studies and are usually associated with com-
positional heterogeneity of the PNRs. The random distribu-
tion of B-site cations creates compositional variations of
local polar nanoregions which in turn lead to locally varying
ferroelectric transition temperatures.
14
Note also that the fre-
quency dependence of the temperature of the dielectric per-
mittivity maximum, T
m
, follows the Vogel-Fulcher law (not
shown here) as reported by us previously.
3
The dielectric permittivity e0of the relaxor ferroelectric
can be described by the Curie Weiss law (Eq. (1)) in the high
temperature paraelectric phase
1
e0¼Th
C:(1)
To determine the critical parameters C (Curie constant)
and the transition temperature h, the higher temperature data
for the dielectric constant e0was fitted to Eq. (1) for all the
BZTx compositions. The fitted data are shown in Fig. 2that
yield the values of Curie constant Cand Curie temperature
h.For example, for the composition BZT0.5 the Curie con-
stant C is 3.33 10
4
K while his 91 K, consistent with val-
ues reported previously.
10
Note that the deviation of the data
from the Curie-Weiss behavior initiates at a temperatures
T
dev
higher than the peak temperature T
m
, as marked in
Fig. 2for BZT0.6.
It has been argued
8,14
that the relaxor ferroelectrics may
be considered as electric dipole glass analog of magnetic
spin glass systems, and therefore relaxor ferroelectrics may
be treated and analyzed using the well-established models
such as the Edwards-Anderson model developed for the spin
glass systems. Hence to discuss the development of the
relaxor state we take into account the EA-order parameter
which in the present context describes the average correla-
tions between the different PNRs. The order parameter qEA
can be written as qEA hPiPji,
3,15,16
where Piand Pjare the
dipole moments corresponding to the ith and jth polar nano-
regions, respectively. Sherrington and Kirkpatrick (SK)
developed
17
an infinite range model for the spin glass to cal-
culate the EA-parameter. This model relates the temperature
dependence of the susceptibility (v) to the local order param-
eter qEA (Refs. 16–19)
v¼C1qEA
ðÞ
Th1qEA
ðÞ
:(2)
FIG. 1. The real and imaginary parts
of dielectric permittivity measured at
different frequencies as a function of
temperature for various compositions
BZTx (0.3 x0.6).
FIG. 2. The inverse of the dielectric permittivity (1/e0) as function of temper-
ature for various concentrations, for f¼500 kHz. The lines are the fit to the
Curie-Weiss behavior. The deviation temperature T
dev
and the x-intercept T
o
are marked for the BZT0.6.
262905-2 Usman et al. Appl. Phys. Lett. 103, 262905 (2013)
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On: Tue, 31 Dec 2013 15:27:52
This relation has been used extensively
16,19
to extract
the EA-order parameter from the experimental data for relax-
ors. For this purpose the values of C and has determined
from the Curie Weiss fit of the data and the value of dielec-
tric susceptibility (at a particular frequency) for varying tem-
perature are inserted in Eq. (2).
Using the data of Figs. 1and 2the value of the EA-order
parameter was calculated for different concentrations of Zr
at a fixed frequency of 500 kHz. The results are shown in
Fig. 3. It is evident from the figure that for all concentrations
of Zr, qEA T
ðÞ
starts smoothly and slowly at high tempera-
tures and then rises rapidly and appears to merge at low
enough temperatures. For each composition the value of qEA
becomes nonzero at a temperature closely coinciding with
the temperature T
dev
defined earlier as the temperature where
the 1/e0vs. Tdata begin to deviate from the Currie–Weiss fit
(see Fig. 2). In general, we observe that the onset point for
nonzero qEA value shifts to lower temperatures with increas-
ing x. Note that at fixed temperature, e.g., at 100 K, qEA T
ðÞ
for higher concentrations is lower than qEA T
ðÞ
at low con-
centrations. All these observations are consistent with the
PNR or polar clusters picture.
11
For example, at low Zr con-
centration paraelectric BaZrO
3
forms a dilute solution in fer-
roelectric BaTiO
3
. At this stage the polar clusters that are
centered at Ti
þ4
sites are in close proximity to each other
and are able to develop strong correlations even at relatively
higher temperatures. On the other hand increasing Zr con-
centration results in a lower number of Ti-centers in the BZT
solid solution and hence to poor correlations among the
PNRs at higher temperatures. These correlations and the
EA-order parameter qEA T
ðÞtherefore grow rapidly as the
temperature is reduced since more and more PNRs forms
and smaller PNRs merge to form even bigger clusters at low
temperatures. Finally, at low enough temperatures the num-
ber of PNRs begins to saturate and the order parameters
qEA T
ðÞfor the respective concentrations merge at low
enough T.
In order to determine if the behavior of the order parame-
ter satisfies any universal relation, we plotted the qEA values
for different concentrations vs. their respective scaled temper-
atures T/T
m
. The data are shown in Fig. 4. Interestingly we
found that the qEA T=Tm
ðÞ
curves for different concentrations
(but the same frequency) follow a universal pattern. This is
evident from Fig. 4where the curves for the four different
concentrations overlap to a very high degree up to
T0.85 T
m
. Above this temperature range all the curves
show a clear tail with qEA vanishing at Tsignificantly higher
than T
m
. The non-vanishing of qEA at T¼T
m
and the devia-
tion from the extrapolated fit at higher temperatures is not
surprising since the ordered regions or PNRs are understood
to exist at temperature well beyond the Curie point, up to the
Burns temperature.
10
However their numbers are too few, or
the interactions between them are too weak to develop a cor-
related behavior.
Typically spin glass systems have been shown to follow
a temperature dependence of universal sort given as
1T=Tg
n, where the value of ncan vary depending on
the dimensionality of the system, the assumed form of the
random field distribution, and the proximity to the critical
temperature.
19,20
In our case we found that the qEA T=Tm
ðÞ
dependence that most closely describes the universal behav-
ior as shown by the solid line in Fig. 4is given by the de-
pendence qEA T=Tm
ðÞ
¼1T=Tm
ðÞ
nwith n ¼2.0 60.1.
While such a dependence is not typical for spin glass sys-
tems, it has been observed in an unconventional spin glass
Mg
1þt
Ti
t
Fe
22t
O
4
.
21
At this point the scaling behavior that
we observe can only be justified as an empirical fit.
The observed behavior of qEA has also been compared to
the predictions of the spherical random bond-random field
(SRBRF) model which has been applied to heterovalent
relaxors.
22
First this model predicts a linear Tdependence
for qEA at low temperatures, if one is to ignore random fields.
We however do not find linear qEA(T/T
m
) dependence for any
appreciable temperature range. Second, if random fields are
considered as playing a significant role, then the SRBRF
model predicts a systematic change in the qEA (T/T
m
) behav-
ior with increasing random field strength, i.e., the absence of
FIG. 3. The EA order parameter qEA as a function of temperature for various
BZTx compositions and f¼500 kHz determined using Eq. (2). Note that the
onset temperature T
dev
of qEA is the same as the deviation temperature deter-
mined from the Curie-Weiss behavior.
FIG. 4. EA-order parameter qEA as a function of scaled temperature T/T
m
for
various BZTx compositions and f¼500 kHz. The degree of overlap of the
curves below T¼0.85 T
m
and the deviations at higher Tare evident. The solid
curve indicates the scaling function qEA ¼1T=Tm
ðÞ
nwith n¼2.05 60.1.
262905-3 Usman et al. Appl. Phys. Lett. 103, 262905 (2013)
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a universal scaled behavior. We, on the other hand, find an
overlapping or universal qEA(T/T
m
) behavior for the various
compositions. We understand the higher xcompositions as
corresponding to higher level of random field strengths D.
22
Thus if random fields were to be playing a significant role in
determining the order in these relaxors than the qEA(T/T
m
)
curves for different x should not overlap. Therefore we
emphasize that our data show that in case of homovalent sub-
stituted relaxors, the role of random fields is not substantial.
This is also in agreement with the results of Refs. 11, where
they found that the relaxor behavior continues in BZT0.5
even if they completely switched off the random fields in
their simulations. They have argued that small antiferroelec-
tric contributions present in this system may initiate the
observed relaxor behavior.
Finally we have addressed the question of the dynamical
behavior of the order parameter by comparing the qEA (T/T
m
)
behavior over a wide range of frequencies. This behavior has
been studied for all the four compositions (Fig. 5). It is appa-
rent from Fig. 5(a) that qEA(T/T
m
,f) shows a frequency inde-
pendent behavior for x¼0.3. We note that this is the
composition where the relaxor behavior is known to just set
in for the BZT system. This may also be understood as the
concentration where polar cluster sizes grow large enough
and correlations set in between them. However as xincreases
we note that the scaled qEA(T/T
m
,f) demonstrates significant
frequency dependence, as evident from Fig. 5. We see that
the high frequency curves lie systematically below the low
frequency curves. In other words, at the same scaled temper-
ature smaller order parameter qEA(T/T
m
,f) values are associ-
ated with the higher frequencies suggesting obviously that
the degree of correlation decreases with increasing fre-
quency. Note also that as we move to higher concentrations
(Figs. 5(b)–5(d)) and thereby deeper into the relaxor state,
the gap between the low frequency and the high frequency
curves increases. This can be interpreted in the sense that for
larger xcompositions there is a broader distribution of relax-
ation times which is reflected in a greater level of disper-
sion.
23
We also note that for a fixed xthe onset point, where
qEA(T/T
m
,f) initially assumes a non-zero value, shifts to
lower scaled temperatures with increasing frequencies. This
indicates that with increasing frequency less time is available
for the correlations to develop and therefore low values of
qEA(T/T
m
,f).
In conclusion we have presented a detailed study of BZT
relaxor ferroelectrics with the view to relate the dielectric
response of this system with the picture of a dipolar glassy
system. The behavior is seen to be well described by the de-
velopment of an order parameter via mean field theory. For
fixed frequency a universal behavior of the order parameter
is reported for a range of relaxor concentrations. This univer-
sal behavior also supports the argument that in these isova-
lent substituted relaxor systems the random fields may not
play a significant role. The dynamic behavior of the order pa-
rameter is also consistent with a weakening of the correla-
tions between the ordered regions with increasing frequency.
This work was financially supported by the Higher
Education Commission of Pakistan under the grant for the
project Development and Study of Magnetic Nanostructures.
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FIG. 5. EA-order parameter qEA for
various compositions BZTx as function
of reduced temperature T/T
m
measured
at frequencies of 0.2–500 kHz.
262905-4 Usman et al. Appl. Phys. Lett. 103, 262905 (2013)
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On: Tue, 31 Dec 2013 15:27:52
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On: Tue, 31 Dec 2013 15:27:52
Response to “Comment on ‘Order parameter and scaling behavior in
BaZr
x
Ti
12x
O
3
(0.3 <x<0.6) relaxor ferroelectrics’ ”
[Appl. Phys. Lett. 104, 156102 (2014)]
Muhammad Usman, Arif Mumtaz,
a)
Sobia Raoof, and S. K. Hasanain
Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan
(Received 5 March 2014; accepted 3 April 2014; published online 17 April 2014)
[http://dx.doi.org/10.1063/1.4871415]
This reply is in response to a Comment posed by
J. Macutkevic and J. Banys on our recently published Letter
1
as mentioned in the title.
Considering the arguments of Macutkevic and Banys,
2
we explain the physical basis of the equations used in the
analysis of the temperature dependence of the dielectric
response, in terms of the Edwards-Anderson like order
parameter.
3
As discussed by Vugmeister and Rabitz,
4
the Edward-
Anderson order parameter qEA in case of relaxors,
5
where it
relates to a non-equilibrium freezing phenomenon, is a time
dependent quantity. In this case, qEA describes the fraction of
clusters effectively frozen at time t. Using a model that
includes a broad distribution of relaxation time, they obtain
the relation between the dielectric response and qEA (t) as
e0ðx;TÞ¼ keo1qEA
ðÞ
1k1qEA
ðÞ
þeo;(1)
where qEA ¼qEA (x,T) and k(T) is a parameter whose devia-
tion from h/T indicates the deviation from the mean field
picture. This is obviously a result very similar to the suscep-
tibility in the infinite ranged Sherrington-Kirkpatrick model
6
where k¼h/T. Various references exist in magnetic spin
glasses where this expression has been used to analyze data
on different time scales.
7
We have used the above expression
to extract qEA from the e0(x,T) data along the lines of
Viehland et al.
5
who studied the response of lead magnesium
niobate (PMN) at 100 kHz using the same expression.
It is of course correct that in relaxors a long range or-
dered state is never realized. Experimentally, a deviation
from the Curie-Weiss law is observed
5,8
in relaxors at a typi-
cal temperature above the peak. This deviation is again very
commonly explained in terms of the formation of polarized
clusters and interaction between them. It has recently been
shown in numerical simulations of relaxors by Akbarzadeh
et al.
9
that there is a high temperature region where Curie
Weiss law is obeyed and with decreasing temperatures T
dev
indicates the onset of cluster formation while at further lower
temperatures the cluster begin to interact leading to further
deviation from the Curie Weiss law.
1
M. Usman, A. Mumtaz, S. Raoof, and S. K. Hasanain, Appl. Phys. Lett.
103, 262905 (2013).
2
J. Macutkevic and J. Banys, Appl. Phys. Lett. 104, 156102 (2014).
3
S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5(5), 965 (1975).
4
B. E. Vugmeister and H. Rabitz, J. Phys. Chem. Solids 61(2), 261 (2000).
5
D. Viehland, S. J. Jang, L. Eric Cross, and M. Wuttig, Phys. Rev. B 46(13),
8003 (1992).
6
D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35(26), 1792 (1975).
7
K. Binder and A. P. Young, Rev. Mod. Phys. 58(4), 801 (1986); K. Binder,
in Festk€
orperprobleme 17, edited by J. Treusch (Springer, Berlin,
Heidelberg, 1977), Vol. 17, p. 55.
8
V. V. Shvartsman and D. C. Lupascu, J. Am. Ceram. Soc. 95(1), 1 (2012);
A. Samara George, J. Phys.: Condens. Matter 15(9), R367 (2003).
9
A. R. Akbarzadeh, S. Prosandeev, E. J. Walter, A. Al-Barakaty, and L.
Bellaiche, Phys. Rev. Lett. 108(25), 257601 (2012).
a)
Author to whom correspondence should be addressed. Electronic mail:
arif@qau.edu.pk
0003-6951/2014/104(15)/156103/1/$30.00 V
C2014 AIP Publishing LLC104, 156103-1
APPLIED PHYSICS LETTERS 104, 156103 (2014)
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On: Fri, 18 Apr 2014 04:02:30
Comment on “Order parameter and scaling behavior in BaZr
x
Ti
1–x
O
3
(0.3 <x<0.6) relaxor ferroelectrics” [Appl. Phys. Lett. 103, 262905 (2013)]
J. Macutkevic and J. Banys
Faculty of Physics, Vilnius University, Sauletekio 9, Vilnius LT-10222, Lithuania
(Received 18 February 2014; accepted 3 April 2014; published online 17 April 2014)
[http://dx.doi.org/10.1063/1.4871414]
Usman et al.
1
have performed dielectric investiga-
tions of BaZr
x
Ti
1x
O
3
(BZT) ceramics in the frequency
range from 200 Hz to 500 kHz. In this Comment, we
emphasize that the dielectric data analysis in Ref. 1is not
reasonable.
The Curie-Weiss fit
1
0¼TTC
C(1)
does not have physical sense for relaxors. Indeed, according
to the Landau-Ginzburg theory for ferroelectrics, the Curie-
Weiss law is not valid far and very close to the ferroelectric
phase transition temperature.
2
For typical relaxors, no ferro-
electric phase transition occurs without bias electric field.
3
So, for relaxors, it is even not possible to determine in which
temperature range the Curie-Weiss law could be applied.
4
In
fact, the Curie-Weiss fit (Eq. (1)) is the linear approximation
of 1=0ðTÞand could be applied for any materials in some
temperature region. Moreover, the formula
0¼Cð1qEAÞ
THð1qEAÞ(2)
is valid only for the static dielectric permittivity
ð0ð!0ÞÞ.
5,6
Close to temperature of the maximum of
dielectric permittivity (T
m
), independent from the electro-
magnetic wave frequency, the authors of Ref. 1not have the
static dielectric permittivity data for BZT ceramics; there-
fore, it is rather impossible close to T
m
to perform fit with
Eq. (2). For relaxor (as well for BZT), the temperature de-
pendence of dynamic (frequency dependent) dielectric
permittivity can be correctly described only if the distribu-
tion of relaxation times is known
7
e0ðxÞ¼e1þDeð1
1
fðsÞdðlnsÞ
1þx2s2;(3a)
e00ðxÞ¼Deð1
1
xsfðsÞdðlnsÞ
1þx2s2:(3b)
It was demonstrated by means of Nuclear Magnetic Resonance
(NMR) that below 300 K temperature the Edvards-Anderson
order parameter increases nearly linearly with decreasing tem-
perature in typical relaxor Pb(Mg
1=3
Nb
2=3
)O
3
(PMN).
8
Therefore, the conclusions that the Edvards-Anderson order
parameter is dependent from temperature as q
EA
¼1(T/T
m
)
n
have a saturation in temperature dependence close to T
m
and
can be dependent from the electromagnetic wave frequency
for BZT ceramics are not reasonable.
This research was funded by European Social Fund
under the Global Grant measure.
1
M. Usman, A. Mumtaz, S. Raoof, and S. K. Hasanain, Appl. Phys. Lett.
103, 262905 (2013).
2
L. Landau, Zh. Eksp. Teor. Fiz. 7, 19 (1937).
3
N. de Mathan, E. Husson, G. Calvarin, A. W. Hewat, and A. Morell,
J. Phys. Condens. Matter 3, 8159 (1991).
4
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084104 (2010).
5
D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975).
6
R. Pirc and R. Blinc, Phys. Rev. B 60, 13470 (1999).
7
J. Macutkevic, S. Kamba, J. Banys, A. Brilingas, A. Pashkin, J. Petzelt, K.
Bormanis, and A. Sternberg, Phys. Rev. B 74, 104106 (2006).
8
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Levstik, and R. Pirc, Phys. Rev. Lett. 83, 424 (1999).
0003-6951/2014/104(15)/156102/1/$30.00 V
C2014 AIP Publishing LLC104, 156102-1
APPLIED PHYSICS LETTERS 104, 156102 (2014)
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