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The effects of grain and particle size on the microwave dielectric properties of ferroelectric barium titanate

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The Graduate School
THE EFFECTS OF GRAIN AND PARTICLE SIZE ON THE MICROWAVE
DIELECTRIC PROPERTIES OF FERROELECTRIC BARIUM TITANATE
A Thesis in
Materials
by
Mark Patrick McNeal
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
December 1997
I
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UMI Number:
9817532
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We approve the thesis of Mark Patrick McNeal.
Date of Signature
Robert E. Newnham
Alcoa Professor of Solid State Science
Thesis Co—Adviser
Chair of Committee
( f , / f f 'p
Sei-Joo Jam
Senior Resqpfch Asso6fate
Associate Professor o f Materials
Thesis Co-Adviser
L. Erie'Cross
Evan Pugh Professor of Electrical
Engineering
O dt 9 )9 q i
(& a A £ > , / 9 9 7
Amar S. Bhalla
Senior Scientist and Professor of Materials
and Electrical Engineering
Lynfi A. Carpente
Associate Professor of Electrical
Engineering
CJofRobert N. Pangbom
Professor of Engineering Mechanics
Chair of the Intercollege Graduate
Program in Materials
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g-,
7
ABSTRACT
There is an increasing need for ferroelectric materials for use in various RF and
microwave systems. Potential applications include high frequency decoupling capacitors,
microwave absorber materials, and voltage tunable phase shifter materials. In order that the
full potential of ferroelectric materials is realized, their high frequency properties must be
better understood. It has been shown that ferroelectric polycrystalline ceramic materials
exhibit a large dielectric relaxation, characterized by a decrease in the dielectric constant and
a peak in the dielectric loss. Mechanisms attributed to the relaxation phenomenon include
piezoelectric resonance of grains and domains, inertia to domain wall movement, and the
emission of gigahertz shear waves from ferroelastic domain walls, consequently, the
relaxation phenomenon is intimately linked to the domain state of the ferroelectric. This
being the case, the potential to tune the high frequency properties of ferroelectrics through
control of the domain state exists, thereby making these materials more suitable for device
implementation. One way o f modulating the domain structure o f ferroelectrics is by
shrinking the crystallite size. The focus of this work was to investigate the microwave
dielectric properties of ferroelectric barium titanate as a function of grain and particle size.
In this work, polycrystalline ceramic ferroelectric B aT i03 and BaTiO, powder-polymer
matrix composites were employed. The composite samples were used to decouple
resonances between adjacent grains as well as reduce the three dimensional clamping
experienced by grains in a ceramic. Characterization studies were performed to determine
the effects of grain size and particle size on the lattice parameters, tetragonality, and domain
structure. High frequency (microwave) dielectric properties were carried out using various
techniques (lumped impedance, cavity perturbation and dielectric post resonance) to
identify and correlate relaxation mechanisms with the material characterization results.
Ceramics possessing average grain sizes of 14.4 (coarse grain), 2.14 (small grain), and
0.26 Jim (fine grain) and composites possessing average particle sizes of 1.33 pm, 0.19
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
pm, and ~ 66 nm, were used in this study. The dielectric spectra of the ceramic and
composite samples were measured through microwave frequencies. All samples exhibited
evidence o f relaxation or resonance in their dielectric spectra. The coarse grain ceramic
exhibited Debye—like relaxation with a AK of ~ 1700 and a relaxation frequency o f 771
MHz. Using relaxation models, a domain size of 0.98 pm and a Ps of 32 pC/cm2 were
calculated from the measured relaxation parameters. The small grain ceramic material
exhibited the highest K and loss tangents through all measured frequencies, compared to
the other materials. These observations were attributed to the high density o f 90° domain
twinning associated with stress relief at this grain size. Measurements up to 200 MHz
revealed the onset of the relaxation phenomenon at ~ 100 kHz. The dielectric spectrum of
the fine grain ceramic exhibited distinct resonant character above 400 MHz, as well as the
lowest losses o f all ceramic materials up to ~ 3 GHz. The resonant behavior o f the
dielectric spectrum was attributed to the coupled resonance o f piezoelectric single domain
grains.
The high frequency properties varied more systematically with particle size. The TAM
(1.33 pm) composite exhibited Debye-like relaxation with a AK o f ~ 10 and a relaxation
frequency o f 1.5 GHz. The BT-8 (0.19 pm ) composite also exhibited relaxation in its
dielectric properties. The relaxation frequency, however, was clearly shifted with a
corresponding decrease and broadening o f the loss peak, compared to that of the TAM
composite. The relaxation frequency was determined to be ~ 3.2 GHz. Using the
relaxation models, a domain size of 0.48 and 0.23 pm was calculated for the TAM
composite and BT—8 composite, respectively. The BT-16 composite exhibited a clearly
lower K at all frequencies, as well as the lowest high frequency losses up to ~ 5 GHz. At
2.5 GHz the K and loss began to show an increase, possibly due to the onset o f some
resonance phenomenon. A possible explanation is the existence o f single domain particles
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behaving as independent piezoelectric resonators. By using a dielectric mixing model, the
K of the ferroelectric particles was calculated. From the measured AK of the TAM
particles, a Ps o f 25 p.C/cm2 was calculated.
Finally, the domain size dependence on grain/particle size was examined by using the
calculated domain widths o f the coarse grain ceramic, the TAM composite and the BT—8
composite. The trend in domain size with decreasing grain/particle size was fitted to a
power law. The power law was used to determine the size at which the particle size and
domain width are equivalent. The size was determined to be 0.265 pm.
This work clearly showed the potential to tune the microwave properties o f ferroelectrics
through control of the domain state. In general, relaxation frequencies increased and loss
tangents decreased with decreasing grain/particle size. Relaxation frequencies were
generally governed by the smallest resonant width, i.e., the domain width.
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vi
TABLE OF CONTENTS
LIST OF FIGURES
.......................................................................................................ix
LIST OF TABLES
........................................................................................................ xiii
ACKNOWLEDGMENTS
........................................................................................... xvi
CHAPTER I INTRODUCTION
1.1 Microwave Engineering
1.2 Ferroelectric Applications
CHAPTER B BACKGROUND
............................................................................ 1
............................................................... 1
2
............................................................................ 11
............................................................................ 11
2.1 Dielectric Properties
2.2 Theory of Barium Titanate
................................................................ 21
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
2.2.6
Crystal Structure and Symmetry
...................................... 21
Ferroelectricity
26
....................................................35
Electromechanical Coupling
Domains
.............................................................................. 38
Properties o f Ferroelectrics
................................................... 41
Size Effects
..............................................................................44
2.2.6.1 Particle Size Effects
44
2.2.6.2 Grain Size Effects
................................................... 48
CHAPTER m MICROWAVE DIELECTRIC MEASUREMENT
TECHNIQUES
..........................................................................................56
3.1 Microwaves
........................................................................................... 56
3.2 Transmission Line Theory
57
3.2.1 Permittivity Measurements Using Lumped Elements ........... 68
3.2.2 Error Considerations ................................................................ 70
3.3 Wave Propagation Along Circular Cylindrical Waveguides
............ 71
3.3.1 Circular Waveguide Cavity
....................................................79
3.3.2 Quality Factor (0 ) and Field Configuration of T M ^
Mode
82
3.3.3 The Cavity Perturbation Method
..................................... 86
3.3.4 Shorted Dielectric Post Resonator
......................................89
3.3.5 Filled Coaxial Lines and Waveguides
92
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vii
95
3.4 Summary of Measurement Techniques
CHAPTER IV REVIEW OF LITERATURE
.................................................. 97
4.1 Theories for Dielectric Relaxation in Ferroelectrics
........................97
4.2 High-Frequency Relaxation in Ferroelectric Ceramics ........................ 107
4.3 High-Frequency Relaxation in Polydomain Single Crystal
BaTiO,
119
4.4 High-Frequency Relaxation in Singledomain and Paraelectrie
Single Crystal BaTiOj
120
CHAPTER V OBJECTIVE
.......................................................................................... 123
CHAPTER VI EXPERIMENTAL PROCEDURE
.................................................. 126
6.1 Starting Powders
126
..................................................
6.1.1 Powder Characterization
6.1.1.1 Specific Surface Area and Particle Size
Distribution
...............................................................
6.1.1.2 Thermal Analysis
..................................................
..................................................
6.1.13 X -R ay Diffraction
6.2 Ceramic Processing
............................................................................
6.3 Composite Processing
............................................................................
6.4 Characterization of Ceramics and Composites Using Scanning
Electron Microscopy (SEM) ...............................................................
6.5 Characterization of Hydrothermal Powders Using Transmission
Electron Microscopy (TEM) ...............................................................
6.6 Low Frequency Dielectric Property Measurements
........................
6.7 Microwave Frequency Measurements
..................................................
CHAPTER VH RESULTS AND DISCUSSION
127
127
128
129
130
130
132
133
133
134
.................................................. 141
7.1 Ceramic Characterization and Dielectric Spectroscopy
..................... 141
7.1.1 Ceramic Microstructural Characterization
......................
7.1.2 Ceramic X -R ay Diffraction Analysis ...................................
7.1.3 Ceramic Temperature Dependent Dielectric
M easurem ents............................................................................
7.1.4 Ferroelectric Hysteretic Properties
...................................
7.1.5 Dielectric Spectroscopy of Ceramic B aT i03 Samples .........
141
145
148
152
154
7.2 BaTiOj Powder and Composite Characterization and Dielectric
Spectroscopy
174
7.2.1 Particle Size Distributions and Specific Surface Area
Measurements
174
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viii
7.2.2 Powder X -R ay Diffraction
..................................................
7.2.3 Powder Microstructural Characterization Using TEM ...........
7.2.4 Composite Microstructural Characterization Using
SEM ..........................................................................................
7.2.5 Dielectric Spectroscopy o f Composite B aT i03 Powder—
Polymer Composites ................................................................
7.3 Domain Size Correlation
CHAPTER V m SUMMARY AND FUTURE W ORK
8.1 Summary
8.2 Future Work
REFERENCES
176
178
181
183
205
....................................... 207
207
............................................................................................ 212
......................................................................................................... 215
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LIST OF FIGURES
Figure 1.1. The electromagnetic spectrum (Pozar,
1990)
..................................... 2
Figure 1.2. Beam—steering concept using phase shifters at each radiating
element
.....................................................................................................................4
Figure 1.3. Waveguide flange with coaxial high voltage attachment
(Collier, 1992)
....................................................................................................... 5
Figure 1.4. Microstrip phase array antenna
................................................................7
Figure 2.1. Illustration o f electric dipoles crossing surface Asv (a) Negative
charge crossing As, flowing into Av. (b) Positive charge crossing As, flowing
out o f Av
..................................................................................................................... 15
Figure 2.2. (a) Behavior o f dielectric parameters near resonance frequency, (b)
Behavior o f dielectric parameters showing Debye relaxation ....................................... 20
Figure 2.3. Frequency dependence of various polarization mechanisms
Figure 2.4. Cubic perovskite structure
.............21
..................................................................23
Figure 2.5. Unit cell distortions o f B aT i03 polymorphs
....................................... 24
Figure 2.6. Model for the calculation of the locally acting internal field
.............27
Figure 2.7. Free energy functions at various temperatures for a ferroelectric
undergoing a first-order transition
.............................................................................. 32
Figure 2.8. Properties as a function of temperature for a first-order
ferroelectric
...................................................................................................................... 33
Figure 2.9. Static walls separating 180° and 90° domains
....................................... 40
Figure 2.10. (a) Spontaneous polarization versus temperature; and (b) dielectric
constant measured along the a and c directions versus temperature
(Merz, 1949, 1953)
42
Figure 2.11. Dielectric hysteresis loop................. ........................................................... 43
Figure 2.12. Room temperature tetragonality change with particle size in B aT i0 3
powder (Uchino et al., 1990)
Figure 2.13. Particle and grain size dependence of the Curie temperature in
B aT i03 (Uchino et al., 1990)
47
48
Figure 2.14. Temperature dependence of dielectric constant in various grain size,
high purity B aT i03 ceramics (Kinoshita and Yamaji,1976) .......................................49
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Figure 2.15. (a) Deformation introduced in cubic grain due to spontaneous
polarization; (b) internal elastic energy minimization through 90° twinning ............ 52
Figure 2.16. Domain width of B aTi03 as a function of grain
size (Arlt, 1990)
55
Figure 2.17. Change in dielectric constant (relative permittivity, er) with grain size
for B aT i03 ceramics (Arlt, 1990)
55
Figure 3.1. Two conductor transmission line illustrating electric and magnetic
fields associated with TEM mode
.............................................................................. 58
Figure 3.2. Voltage and current definitions and equivalent circuit for an incremental
length of transmission line, (a) Voltage and current definitions, (b)
Lumped-element equivalent circuit
.............................................................................. 59
Figure 3.3. Lossless transmission line terminated by ZL
...................................... 64
Figure 3.4. Illustration o f coaxial sample holder and equivalent circuit
Figure 3.5. Geometry o f circular cylindrical waveguide
........... 69
......................................73
Figure 3.6. Geometry o f cylindrical cavity
................................................................ 80
Figure 3.7. Parallel RLC resonant circuit
................................................................ 82
Figure 3.8. Input impedance magnitude versus frequency
......................................84
Figure 3.9. Electric and magnetic fields o f T M ^ mode. Flux density is
proportional to field intensity ........................................................................................... 86
Figure 3.10. TE0I1 field configuration in shorted dielectric rod resonator
............ 91
Figure 3.11. Mode chart for a dielectric rod resonator short-circuited at both ends
(Kobayashi and Tanaka, 1980)
Figure 3.12. Partially filled rectangular and coaxial waveguides
92
........................93
Figure 3.13. Signal flow graph representation o f a two-port network
............ 94
Figure 4.1. Square face o f cubic volume representing paraelectric crystallite (a),
and deformation of cubic crystallite due to development of P0 (b)
101
Figure 4.2. Spontaneously deformed crystallite in the presence of a static electric
field, Zs, and stress field, Ts .......................................................................................... 102
Figure 4.3. Frequency dependence of dielectric constant (1), and dielectric loss (2)
(Poplavko et al., 1969)
......................................................................................... 114
Figure 4.4. Relaxation frequency versus temperature for BaTi03 (Kazaoui et al.,
1991)
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115
Figure 6.1. Illustration o f sample holder used for lumped impedance
measurements
Figure 6.2. Illustration o f cavity and sample configuration used for cavity
perturbation measurements
.......................................................................................... 136
Figure 6.3. Illustration o f post resonance measurement configuration
............ 139
Figure 6.4. Illustration o f rectangular X -band waveguide measurement
setup .................................................................................................................................. 140
Figure 7.1. Photomicrograph of domain structure and grain sizes in coarse grain
polycrystalline B aT i03 (CGBT) ceramic after polishing and etching ......................... 143
Figure 7.2. SEM photomicrograph of small grain polycrystalline B aT i03 (SGBT)
ceramic, (a) Chemically etched sample showing domain relief, (b) Thermally
............................................................................. 144
etched sample showing grain sizes
Figure 7.3. SEM photomicrograph of fine grain polycrystalline B aT i03 (FGBT)
ceramic thermally etched sample showing grain sizes
Figure 7.4. {200} reflections from XRD patterns obtained on the various
ceramics
..................................................................................................................... 147
Figure 7.5. Average grain size effect on the room temperature dielectric properties
of ceramic B aTi03 at 10 kHz
............................................................................ 149
Figure 7.6. Temperature dependence in the dielectric constant o f CGBT
............ 149
Figure 7.7. Temperature dependence in the dielectric constant o f SGBT
............ 140
Figure 7.8. Temperature dependence in the dielectric constant o f FGBT
............ 150
Figure 7.9. Comparison between the temperature dependence o f the dielectric
constant of ceramic B aT i03 samples of different grain size ..................................... 152
Figure 7.10. Dielectric hysteresis measured in (a) CGBT and (b) SGBT and
FGBT
.................................................................................................................... 153
Figure 7.11. Strain versus electric field for CGBT, SGBT, and FGBT
.......... 154
Figure 7.12. Spectrum o f dielectric properties measured from CGBT
ceramics
.................................................................................................................... 164
Figure 7.13. Spectrum o f dielectric properties measured from SGBT
ceramics
.................................................................................................................... 167
Figure 7.14. Spectrum o f dielectric properties measured from FGBT
ceramics
.................................................................................................................... 172
Figure 7.15. Comparison o f dielectric spectra measured from the various
ceramics
.................................................................................................................... 173
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145
Figure 7.16. Particle size distributions of (a) TAM , and (b) BT—8
........................ 175
Figure 7.17. {220} reflections from XRD patterns obtained on the various
powders
...................................................................................................................... 177
Figure 7.18. TEM images o f hydrothermal powders at 200,000 X magnification:
(a) bright field image of BT—8 powder, (b) dark field image of B T-8 powder,
(c) bright field image of BT—16 powder, and (d) dark field image of BT—16
powder........................................................ ......................................................................... 180
Figure 7.19. Microstructure o f TAMC composite
................................................... 182
Figure 7.20. Microstructure o f BT-8C composite
................................................... 182
Figure 7.21. Microstructure of BT-16C
................................................................ 183
Figure 7.22. Spectrum of dielectric properties measured from TAMC
composite.................................................... ......................................................................... 191
Figure 7.23. Spectrum of dielectric properties measured from BT-8C
composite.................................................... .......................................................................... 200
Figure 7.24. Spectrum of dielectric properties measured from B T-16C
.......................................................................... 201
composite
Figure 7.25. Comparison of dielectric spectra o f various composite
samples
........................................................................ 202
Figure 7.26. Dielectric spectrum of TAM particles using mixing rule
Figure 7.27.
Correlation between domain width and grain/particlesize
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204
............206
LIST OF TABLES
Table 1.1 Typical Frequencies and Approximate Band Designations
........... 2
Table 2.1 Names, SI Values and Units of Important Electromagnetic
Quantities
..................................................................................................................... 12
Table 2.2 Measured Values o f d^
............................................................................. 46
Table 3.1 Values o f p '^ for TEm Modes of a Circular Waveguide
........................ 77
Table 3.2 Values o f p ^ for T M ^ Modes of a Circular Waveguide
........................ 79
Table 3.3 Summary of Microwave Dielectric Measurement Techniques
...........96
Table 4.1 Summary of Reported Dielectric Properties o f Ceramic B aT i03 Below
and Above the Relaxation Frequency
Table 6.1 Commercial Powder Sources
118
................................................................ 126
Table 7.1 Measured Densities of BaTi03 Ceramic Samples
142
Table 7.2 Grain Size Statistics of BaTi03 Ceramic Samples
145
Table 7.3 Summary of Calculated Lattice Parameters
146
Table 7.4 Ceramic Room Temperature Dielectric Data Obtained at 10 kHz
........... 148
Table 7.5 Summary of Transition Parameters in Ceramic B aT i03 Samples ........... 151
Table 7.6 Room Temperature Low Frequency Dielectric Properties of
CGBT
155
Table 7.7 Representative Raw Data and Calculated K and tan8 From Lumped
Impedance Measurements Conducted on CGBT
................................................... 156
Table 7.8 Room Temperature Lumped Impedance Measurements on
CGBT
157
Table 7.9 Measured Parameters and Calculated Dielectric Properties o f CGBT
Using Cavity Perturbation at 1.5 GHz
159
Table 7.10 Measured Parameters and Calculated Dielectric Properties of CGBT
Using Cavity Perturbation at 3 GHz
160
Table 7.11 Measured Parameters and Calculated Dielectric Properties of CGBT
Using Cavity Perturbation at 5.6 GHz
161
Table 7.12 Room Temperature Cavity Perturbation Measurements on
CGBT
162
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xiv
Table 7.13 Room Temperature Low Frequency Dielectric Properties of
SGBT
.................................................................................................
165
Table 7.14 Room Temperature Lumped Impedance Measurements on
SGBT
166
Table 7.15 Room Temperature Low Frequency Dielectric Properties of
FGBT
168
Table 7.16 Room Temperature Lumped Impedance Measurements on
FGBT
169
Table 7.17 Room Temperature Cavity Perturbation Measurements on
FGBT
169
Table 7.18 Summary of PSD and Specific Surface Area Measurements
Table 7.19 Summary of Calculated Lattice Parameters
.......... 176
..................................... 176
Table 7.20 Measured Densities of B aT i03 Powder-Polymer Composites
.......... 181
Table 7.21 Microwave Properties o f Polypropylene
184
Table 7.22 Room Temperature Low Frequency Dielectric Properties of
TAMC
184
Table 7.23 Representative Raw Data and Calculated K and tan8 From Lumped
Impedance Measurements Conducted on TAMC
.................................................. 185
Table 7.24 Room Temperature Lumped Impedance Measurements on
TAMC
186
Table 7.25 Measured Parameters and Calculated Dielectric Properties of TAMC
Using Cavity Perturbation at 1.5 GHz
187
Table 7.26 Measured Parameters and Calculated Dielectric Properties of TAMC
Using Cavity Perturbation at 3 GHz
188
Table 7.27 Measured Parameters and Calculated Dielectric Properties of TAMC
Using Cavity Perturbation at 5.6 GHz
189
Table 7.28 Room Temperature Cavity Perturbation Measurements on
TAMC
190
Table 7.29 Room Temperature Low Frequency Dielectric Properties on
BT-8C
192
Table 7.30 Room Temperature Lumped Impedance Measurements on
BT-8C
192
Table 7.31 Measured Parameters and Calculated Dielectric Properties of BT-8C
Using Cavity Perturbation at 1.5 GHz
194
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Table 7.32 Measured Parameters and Calculated Dielectric Properties of BT-8C
Using Cavity Perturbation at 3 GHz
195
Table 7.33 Measured Parameters and Calculated Dielectric Properties of BT—8C
Using Cavity Perturbation at 5.6 GHz
196
Table 7.34 Room Temperature Cavity Perturbation Measurements on
BT-8C
197
Table 7.35 Room Temperature Low Frequency Dielectric Properties of
.................................................................................................................... 198
BT-16C
Table 7.36 Room Temperature Post Resonance Measurements on
BT-16C
.................................................................................................................... 198
Table 7.37 Summary of Calculated Domain Widths ....................................................205
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xvi
ACKNOWLEDGMENTS
I would like to thank my advisors, Drs. Robert E. Newnham and Sei-Joo Jang, for
their guidance and help throughout this study. It was through their wisdom that this
research endeavor was realized and through their support that it was accomplished. Sincere
appreciation and gratitude are also extended to Drs. L. E. Cross, Amar S. Bhalla, and Lynn
A. Carpenter for their participation on my committee, and their invaluable insights and
suggestions. Thanks are also due to the technical and support staff, here at the Materials
Research Laboratory (MRL), especially to Beth Jones and Dick Brenneman, for their help
and assistance with my research. I would also like to thank Paul Rerhig for supplying the
fine grain ceramics employed in this study.
I am in debt to my friends at MRL as well, those who assisted me wherever they could.
In particular, I am grateful to Dr. Seung-Eek “Eagle” Park for his help and friendship. I
would also like to extend my gratitude to my family, especially my mother, who supported
me throughout my education. A special thanks is in order for my wife Kelley, whom
through her patient support, helped me to complete this work. I cherish and love her very
much. I would also like to acknowledge the financial support received from the National
Science Foundation (NSF), contract DMR 9223847, Size Effects in Ferroics. Finally, I
would like to thank God, without whom, this work would not have been possible.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER I
INTRODUCTION
1.1 Microwave Engineering
Because o f the relatively small wavelengths associated with microwave signals,
standard circuit theory is inadequate in describing microwave networks and circuits. In
describing and understanding the behavior of microwave networks, general field theory
must be used, whereby M axwell’s equations and their solutions, provide a complete
description o f the electromagnetic fields at every point in space. It is for this reason that the
solution to microwave network problems may be challenging; however, in spite of the
complexities which may arise in solving microwave network problems, the benefits of
microwave frequencies make for attractive engineering applications. As an example,
antenna gain is increased at higher frequencies, since antenna gain is proportional to the
electrical size o f the antenna. Also, more bandwidth and therefore, more informationcarrying capacity, can be realized at higher frequencies. Furthermore, line—of-sight travel
of microwave signals make possible communication links with both terrestrial and orbiting
satellites (lower frequency signals are bent by the ionosphere and millimeter wave
frequencies may be highly attenuated by the atmosphere or rain). In addition, the radar
(radio detecting and ranging) cross-section of a radar target is usually proportional to the
target’s electrical size. This fact, coupled with the characteristics o f increased antenna gain,
often make microwave frequencies the preferred band for radar applications. Finally,
various molecular, atomic, and nuclear resonances occur at microwave frequencies, making
microwave characterization an invaluable tool for basic scientific investigations. Figure 1.1
shows the electromagnetic spectrum and Table 1.1 shows typical frequencies and band
designations (Pozar, 1990).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Frequency (Hz)
!l J! 1
11
r
10
I02
1
1
11
!
i
i
I
i i 1 11
1
10
1
I0 ~ '
I0 -2
10 3
10"*
*3
£a*
1
10' 5
Visible light
*=3
3
M icrow aves
' :!ii
!?!!!!
Far Infrared
>
i
3 x I07 3 x |0 8 3 x 109 3 x I 0 10 3 x I 0 11 3 x I0 | : 3 x | 0 13 3 x | 0 14
J_________ I__________1__________ I_________ 1------------- I -------------- L
1
1 jo 1 1
1
i
i
|
Shortwave
radio
V
3 x I06
AM broadcast
radio
3 x 10s
L
10 6
W avelength (m)
Figure 1.1. The electromagnetic spectrum (Pozar, 1990)
Table 1.1 Typical Frequencies and Approximate Band Designations
Typical Frequencies
Approximate Band Designations
AM broadcast band
535-1605 kHz
L-band
1-2 GHz
Shortwave radio
3-30 MHz
S-band
2-4 GHz
FM broadcast
88-108 M Hz
C-band
4-8 GHz
VHF TV (2-4)
54.72 MHz
X—band
8-12 GHz
VHF TV (5-6)
76-88 MHz
Ku-band
12-18 GHz
UHF TV (7-13)
174-216 MHz
K-band
18-26 GHz
UHF TV (14-83)
470-890 MHz
Ka-band
26-40 GHz
Microwave ovens
2.45 GH z
U-band
40-60 GHz
1.2 Ferroelectric Applications
Currently, the majority of applications of microwaves are related to high speed
microelectronics, radar, and communication systems. The requirement for faster and
greater quantity of data transmission has led to the development of high speed computers.
High speed computers use narrow pulses for operation, and with increased computer
speeds, the pulses are becoming narrower, with time scales that fall within the microwave
frequency range. Multilayer ceramic (MLC) substrates support the circuitry for the transfer
of information from a silicon chip to other functions of the computer. Inductance due to
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line discontinuities can cause switching noise and signal distortion, which may result in
computation errors. Decoupling capacitors are used to neutralize line inductance, whereby
decoupling capacitors are placed near the silicon chip to reduce spurious switching in signal
lines. High dielectric constant ferroelectric materials will assist in the material selection for
decoupling capacitors currently utilized in computer packaging. High permittivity, low loss
materials are required to meet the current demands o f small size and high speed; operational
frequencies currently near 1 GHz have already been realized and are continually moving
higher (Lanagan, 1987; Dube and Jang, 1990; and Pozar, 1990).
Interest in the area of ferroelectric materials for non-volatile, high speed random access
memories (FRAMs) and dynamic random access memories (DRAMs) has also increased in
recent years. Ferroelectric thin films (with lead zirconate titanate (PZT) receiving
considerable attention) deposited via sol-gel or sputtering, offer the advantage of high
polarizations which lead to high charge storage densities (Obhi and Patel, 1994; Scott et al.,
1995; and Jones et al., 1994)
Radar systems are used for detecting and locating air, ground, or sea—going targets, by
airport traffic-control radars, missile tracking radars, fire-control radars, and other
weapons systems. The traditional rotating reflector antenna associated with airport traffic
control systems is now being replaced with a more recent design concept called the phased
array. In conventional rotating antennas, a heavy antenna rotating mechanism is needed to
scan the beam. This design configuration is too slow and expensive; the size and weight of
such systems are severe disadvantages in applications like air-born systems. The phase
array concept involves the planar array of possibly thousands o f closely spaced, individual
antenna radiators whose composite beam can be shaped and spatially directed in
microseconds. Beam steering is accomplished by varying the relative phase between
radiating elements, as illustrated in figure 1.2. The phase shifter has the form o f an
electrical delay line, which causes phase shift by controlled variation of the group velocity
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of the microwave signal. Until recently, ferrite phase shifters have been used for phased
arrays because of their high operation speed, weight and size. In a ferrite based delay line,
the shift in group velocity is caused by a change in permeability through variation o f an
applied magnetic field. Unfortunately, the unit cost and complexity o f these systems has
limited them to specialized military applications. To lower the cost, PIN diode phase
shifters were developed, but their high insertion loss limited their use.
Beam Direction
Equiphase Front
2n
sin0,
Radiators NY/
Phase Shifters
Power
Distribution
Network
rP —
Antenna Input
Figure 1.2. Beam-steering concept using phase shifters at each radiating element.
A ferroelectric phase shifter design has more recently been proposed (Elmer and Jang,
1988; Collier, 1992). The design is based on the variation o f the permittivity of the
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ferroelectric phase shifter by application o f a DC electric field, parallel to the polarization o f
the RF energy, and normal to its direction o f propagation. Variations o f the permittivity
alter the RF propagation velocity. One design configuration involves a short section of
waveguide filled with the bulk ferroelectric material. This would constitute a single element
o f the array antenna configuration. This design was proposed, using a Ku—band flange
filled with barium strontium titanate (BST). Figure 1.3 illustrates the filled waveguide
flange. Additional matching layers are required on either side of the flange to overcome
impedance mismatch between air and the high permittivity ferroelectric, which otherwise
would result in reflection o f nearly all incident signals. The sample is split by a thin
conductive layer between the two halves for lowering the voltage requirement (Collier,
1992).
totvtfw
V'VJJJJJl •
non
Figure 1.3. Waveguide flange with coaxial high voltage attachment (Collier, 1992).
Another design involves the use of a microstrip patch antenna system. This design
allows for easy mounting on to missiles, satellites, or even on to the bum per o f cars as part
j
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6
o f a collision avoidance system. The microstrip phased array antennas can be fabricated by
interconnecting individual patches with microstrip lines, as shown in figure 1.4. The
microstrip lines provide power splitting and phasing capabilities which produce the desired
radiation pattern. Beam steering capabilities are obtained by inserting the phase shifter in
the feed network of each patch. A modular two dimensional array radiator would include
the various power dividing networks and phase shifting controllers with a control algorithm
determining the various phase shifter settings for the desired beam positions. The
operational equations which govern the change in phase, A<j>, for a microstrip transmission
delay line are given by,
( 1. 1)
and
a d = 27.3 L
( 1.2 )
In equations (1.1) and (1.2), L is the length of the phase shifter m aterial,/is the frequency
of operation, c is the velocity of light, and £rt and £rfare the two different effective dielectric
constants of the material, and £cff is the effective dielectric constant o f the microstrip line.
The ratio of (1.1) to (1.2) gives the performance criterion for microstrip phase shifters, as
given by equation (1.3),
(degree/dB).
(1.3)
The performance criterion for microstrip phase shifters, given in units o f maximum
phase shift per dB of insertion loss, is dominated by dielectric loss. Consequently, the
future o f low-cost, two-dimensional beam-steerable planar array antennas, utilizing
ferroelectric phase shifters depends on novel material engineering techniques to lower the
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loss tangent to < 0.01, so that insertion loss is maintained on the order o f 1-2 dB. It is
also desirable that the voltage tunability of these materials result in a change of permittivity
o f up to 40-50 %, with application of a suitable DC electric field (Babbitt, et al., 1992; and
Varadan, e ta l., 1992).
Microstrip
Line
Microstrip
Antenna
'E l
IQ
+
Feed Point
A A A
f
— Phase Shifter
A IN
substrate
with
Diamond
Coating
Figure 1.4. Microstrip phase array antenna.
Microwave absorbers with wide absorbance bands are required to eliminate spurious
signals; for example, the elimination of TV “ghost” images due to reflections from
surrounding buildings. Microwave absorbers employing ferrite ceramic plates have been
previously used, however, they are characterized by thick, heavy components, with an
upper frequency use o f ~ 1 GHz. Ferroelectric composites have been proposed as a new
absorbency system for applications above 1 GHz (Chino, 1989 and Jones, 1993). The
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absorber system is characterized by a thickness of ferroelectric powder—rubber composite
backed by a conducting plate. The condition for maximum absorbance is derived by
modeling the absorber as a transmission line terminated by a short circuit, o f material
properties governed by K* and /I*. By matching the impedance of the absorber system to
that o f the incident signal, maximum absorption is obtained. At a particular frequency,
maximum absorbance is determined by the K 'a n d K " of the material, assuming it is
nonmagnetic. Careful selection o f the ferroelectric filler material and tuning of the
composite properties should yield optimum matching conditions (Chino, 1989).
Current trends towards more complexity, higher power, smaller size, lighter weight,
lower cost, and higher frequencies through the integration of transmission lines, passive
components (resistors, capacitors and inductors), and active devices (diodes and
transistors), has led to the development of microwave integated circuits (MICs). There are
two distinct types of MICs. Hybrid MICs have one layer o f metallization for conductors
and transmission lines, with discrete components bonded to the substrate. Monolithic
microwave integrated circuits (MMICs) are a more recent development, where the active
and passive circuit elements are grown or implanted directly on a semiconductor substrate.
These technologies have already progressed to the point where complete microwave
subsystems (receiver front ends, radar transmit/receive modules, etc.) can be integrated on
a single chip a few square millimeters in size.
There is an increasing need for ferroelectric materials for use in these microwave
applications, as just discussed. The use of high—dielectric constant ferroelectric ceramics
and films in most microwave devices requires that they possess low dielectric losses. A
fundamental concern with the implementation of ferroelectric materials in microwave
technologies is that they typically undergo a marked relaxation in their dielectric properties.
This relaxation is characterized by a decrease and saturation of the dielectric constant with
an associated peak in the dielectric loss, with increasing frequency. The high losses, as
i
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previously stated, lead to high insertion losses for the device, and is undesirable in most
high-power applications. The origins o f this relaxation phenomenon has been attributed to
the existence o f domain structures, inherent to ferroelectrics. In order that the full potential
of ferroelectrics is realized in microwave applications, the properties o f ferroelectrics
through microwave frequencies must be better understood. In addition, as
microelectronics, MMICs, and devices continue to shrink in size, smaller, more compact
modules are being developed, incorporating thick and thin film technologies. These
technologies introduce specific microstructural constraints which directly influence the
domain configuration. Consequently, several questions concerning the effects of
microstructure arise and need to be answered; that is, what are the effects of shrinking
crystallite size in at least one dimension on the microwave properties of ferroelectrics?
And, as a more general extension to that question, what are the effects of shrinking
crystallite size in all three dimensions? In addition, what effect does the three dimensional
clamping experienced by grains in a ceramic have on the microwave properties of the
material, and when this clamping is removed, is there an effect on the microwave properties
of ferroelectrics? This work will specifically attempt to measure the microwave properties
of a prototypical ferroelectric, barium titanate (BaTi03). It will also investigate the effects
of three dimensional clamping, as experienced by a grain in a ceramic, as well as
investigate the properties o f a ferroelectric crystallite under more relaxed stress conditions,
and finally, what happens when the crystallite size is reduced. The results of this research
should be applicable to other ferroelectric systems.
This study begins with a discussion introducing the basic concepts o f dielectric
materials. These concepts are then extended to explain the origins and consequences o f
ferroelectricity in B aT i0 3. The origins and dynamics o f domains are presented, and
discussed within the context of shrinking crystallite size under both clamped and free
conditions. Chapter III then goes on to develop several microwave measurement
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techniques suitable for the determination of dielectric properties. Here, fundamental
parameters and terminology used in microwave engineering are introduced, together with
the basic problem-solving algorithm used in the solution of microwave networks. Chapter
IV presents a historical overview of observed high-frequency properties in polycrystalline
ceramic, polydomain single crystals and single domain single crystal BaTi03. Also in this
chapter, relaxation theories which either explained or predicted the high—frequency
properties of ferroelectrics are discussed. Chapter V defines the objectives and scope of
this work, while Chapter VI details the experimental procedure followed in this work by
discussing the characterization techniques used as well as how the microwave
measurements were carried out. Chapter VII presents the results of this study. In this
chapter, the results of the material characterization are presented, as well as the measured
dielectric spectra, through microwave frequencies. The dielectric spectra are discussed
within the context of the observed characterization results, and are related to the relaxation
models presented in Chapter IV. Finally, this work ends with conclusions and suggestions
for future work.
i
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CHAPTER n
BACKGROUND
In order that the high frequency properties of ferroelectric barium titanate (BaTi03) may
be interpreted, it is necessary to have a fundamental understanding o f dielectric materials as
well as an understanding behind the phenomenology leading to the development of
ferroelectricity in B aT i03. Beginning with Maxwell’s equations, this chapter introduces
basic concepts which lead to the development of fundamental dielectric parameters. These
concepts are then extended to explain the origins and consequences of ferroelectricity in
B aT i03. The origins and dynamics of domains are then presented, and finally, discussed
within the context o f shrinking crystallite size under both clamped (ceramic) and free
(particle) conditions.
2.1 Dielectric Properties
Maxwell’s equations are mathematical relations between the electric and magnetic fields,
and their current and charge sources. The four experimentally obtained laws that constitute
Maxwell’s equations are the following:
V •e0E = pv, Gauss’s law for electric field;
(2.1)
V -B = 0, Gauss’s law for magnetic field;
(2.2)
rj B
7 de0E
,
V x — = J h— -9— , Ampere s law;
Ho
d*
(2.3)
Faraday’s law
(2.4)
Table 2.1 gives a list of definitions, values and units for the various parameters used in
equations (2.1) through (2.4).
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12
Table 2.1. Names, SI Values and Units of Important Electromagnetic Quantities
Symbol
L
Name
Henry
Quantity or Value
inductance
Units
magnetic flux/current
V
volt
voltage
work/charge
C
Coulomb
charge
current x time
A
Ampere
current
charge/time
Wb
Weber
magnetic flux
m
meter
length
E
electric field intensity
V/m
B
magnetic flux
density
Wb/m 2
J
current flux density
P,
electric charge
density
C /se c
2
-----r— or A/m
m
C/m 3
eo
permittivity o f free
space
8.854 x l( T 1 2
Po
permeability o f free
space
47tx 1(T7
C2
—;—r- or F/m
N m
N s2
, or H/m
C2
In the above equations, V is a vector differential operator and depends on the coordinate
system being employed. We begin, then, by asserting that all materials consist o f atoms,
which are comprised of subatomic charged particles. In the presence of electric and
magnetic fields, forces are exerted on these particles. Three basic phenomena may result
from this interaction between electromagnetic fields and charged particles in condensed
matter: conduction, polarization, and magnetization. Depending on the predominant
phenomenon, the material is classified as a conductor, dielectric, or magnetic material.
This work concerns itself with dielectric materials. In dielectric materials, bound electrons
prevail and their basic response to the application of an external electric field is their
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displacement relative to other bound charges (nuclei). This displacement is analogous to
the mechanical stretching of a spring, an action that has the effect o f storing energy;
consequently, dielectric materials are classified by their ability to store electric energy.
An electric dipole moment, p , is defined as two point charges, +q and -q, o f equal
magnitude and opposite sign, separated by a small distance, where, p = q d , and d is the
vector separation distance between the charges. The dipole moment describes the
microscopic properties o f the material. Macroscopically, dielectric materials are
characterized by their polarization, defined as the electric dipole moment per unit volume.
By defining the average dipole moment per molecule as pa, and the average vector
separation distance as da, polarization, P is given by,
(2.5)
where n is the number o f dipoles per unit volume, nAv is the number o f dipoles in a
volume Av, and p is the positive charge density. In the presence of a time varying electric
field, E , induced electric dipoles oscillate with the field. This flow o f charge is equivalent
to an induced oscillating current called the polarization current. In a linear dielectric,
polarization P , is linearly proportional to the applied electric field so that,
P=
where e 0 is the permittivity of free space (8.854 x 10' 1 2 C 2/ N • m 2), and
(2.6)
is the electric
susceptibility of the material. The electric susceptibility, %e, describes the ability of the
material to be polarized in the presence o f an electric field. The induced current density, J ,
is given by,
(2.7)
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14
Ampere’s law (equation (2.3)) is modified from its free space form by adding in the
polarization current, as follows:
V x^- = 7+^
m
df
+^ = 7+^
dt
dt
+^
^
dt
= J + |- [ ( x , + 1 )£„£].
dt
(2 .8 )
The quantity K = x e + 1 is called the relative permittivity or dielectric constant. Ampere’s
law now becomes,
V x - = 7 + ^ i
Ho
at
i =
dt
= 7+— ,
dt
(2.9)
where e = e0K is the permittivity of the dielectric medium, and
D = e QKE = P + eQE ,
(2.10)
is the electric flux density or dielectric displacement.
Due to the electric field induced polarization, there may be an induced polarization
charge density inside the material. This is illustrated by considering a small volume
element, Av, bound by the surface element, As,, inside the polarized material. Dipoles
completely contained within the volume element contribute no net charge induced inside the
volume Av. However, dipoles near the surface may contribute a net charge to the
differential volume element. Specifically, only those dipoles which cross the surface, and
whose centers lie within a distance ^ c o s 0 , 0 being the angle between the dipole and
surface normal, contribute a net charge to the volume element. Figure 2.1 illustrates this
condition. Figure 2.1 (a) shows the surface element As, and a volume o f ^ c o s 0 depth
extending into the surrounding material above the surface element. Each of the induced
dipoles that has its center within a distance % c o s 0 above the surface contributes a
negative charge that crosses the element of area As,, into the Av. If n is the number of
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15
dipoles per unit volume, then the number o f dipoles with their centers within a distance
dy^cosO from the surface will be n(d/^cosQ )Asv The number of negative charges that
flow into Av is given by
n(-q) ^cosG C -A ^,).
(a)
(2.11)
As.
—COS0
-C O S 0
Figure 2.1. Illustration of electric dipoles crossing surface As,, (a) Negative charge
crossing As, flowing into Av. (b) Positive charge crossing As, flowing out o f Av.
Similarly, as shown in figure 2.1 (b), each of the induced dipoles that has its center within
a distance ^ c o s © below the surface As,, inside Av, contributes a positive charge leaving
the Av. The total positive charge out flowing from the element of volume is
n(q)
cos0 (As,).
Since an outward flowing positive charge is equivalent to an inward flowing negative
charge, the total increase in the negative charge in the element of volume Av partially
enclosed by As, is
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(2.12)
16
2 (n ^ ^ ^ c o s0 A j|) = nqdcosQAsl.
(2.13)
The total increase in the negative charge density in the volume element Av enclosed by a
surface As is given by the product -ppAv (where pp is the polarization charge density), and
is related to the polarization by
n q d cos 0Ay = P ■Ay = - p pAv .
(2.14)
In a bulk material, the induced polarization charge may be related to the polarization by
subdividing the slab into small volume elements and summing the result o f (2.14) over all
the subvolumes. In the limit where all subvolumes and areas become infinitesimally small,
(2.15)
By applying the divergence theorem to the left-hand side of (2.15),
(2.16)
from which the point form is obtained as
(2.17)
Equation (2.17) says that the net outflow of the polarization flux density at a point is equal
to the net polarization negative charge at this point.
Thus far, scalar quantities, such as electric susceptibility %e, permittivity £, and
dielectric constant AT, have been used. However, these quantities are absolute values or
moduli o f the more general complex quantities, x'e, £*, and K m. The complex dielectric
constant is given by K ' = K ' —jK " , where K ' is the real part (lossless) and K " is the
imaginary part, which accounts for loss (heat) in the dielectric medium due to some finite
conduction and/or damping of the vibrating dipole moments. The ratio %
- = tan 5 is
called the dissipation factor or dielectric loss. It represents the fraction of energy
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17
dissipation or power loss, such that 90°-8 is the phase difference that a current leads an
applied alternating current (ac) voltage across a real dielectric (Iskander, 1992; and
Moulson and Herbert, 1990).
There are four basic mechanisms o f dielectric polarization: space charge, dipolar or
orientational, ionic, and electronic. Space charge polarization involves mobile charges
which are present because they are impeded by interfaces. They are not supplied or
discharged at an electrode, and are therefore trapped in the material. Dipolar polarization is
associated with the presence of permanent electric dipoles which exist even in the absence
o f an electric field. In the absence o f an external field, permanent dipole moments of
individual molecules are randomly oriented and yield no net polarization on a macroscopic
scale. When a field is applied, however, these dipoles attempt to reorient in the direction of
the field, and therefore contribute to the polarization. Ionic polarization involves the
displacement of positive and negative ions relative to one another. Finally, electronic
polarization is associated with the shift of the center of gravity of the negative electron
cloud in relation to the center of the positive atomic nucleus in an electric field. The
frequency dependence of these various polarization mechanisms may exhibit resonance or
relaxation effects.
In treating ionic and electronic polarizations, to a first approximation, the electrons and
ions behave as though bound to equilibrium positions by linear springs so that the restoring
force is proportional to displacement, x. An equation of motion for a bound charge in a
time varying field can be written as,
d 2x
dx
m — + m y— + mco* x = qE0 ex p (ja)t),
dt
dt
(2.18)
where y is a damping factor to account for energy losses, m is mass and (Oq is some natural
angular frequency o f oscillation. The solution to (2.18) yields,
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18
gE0 expQcor)
X(t) =
(2.19)
/n{(0)o —co2) + /y©}
Following (2.5), a complex polarization can be written as
P m= —n q x (t).
(2.20)
By inspection and comparison to (2.6),
=
nq
1
m e0 l(©o —©2) + /y©
( 2 .21 )
so that,
a:: = 1 +
nq
m eQ (©o + ©2) + jyca
( 2 .22 )
From (2.22), the real and imaginary parts are
K L - 1=
©3 - ©
nq'
2
m er ( © o - ©">\"z j + 2y 2©
(2.23)
and
K" =
nq
7©
l ( © „ - © 2) -j-y2 ©
(2.24)
2
Because these resonances typically occur at optical frequencies, the subscript
is used to
distinguish these values from low frequency values. The results of equations (2.23) and
(2.24) are shown in figure 2.2 (a) and illustrate dielectric resonance.
Dielectric relaxation occurs when the polarization processes are relatively slow, is
usually highly temperature dependent and is often connected with diffusional mechanisms.
Space charge and dipolar mechanisms typically exhibit this type o f frequency response. In
this case, the damping and restoring forces are important, but inertial effects (acceleration)
are neglected. The equation o f motion is modified to,
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the solution of which yields,
K ' —K '
K ‘ = 1H— £----- - ,
1 + jayz
K
,
~
1
K ’- K l
= T~~ " 2 2 »
l+ to X
(2.26)
(2.27)
and
K " = (K'S - K L )
1
^
2.
+ co x
(2.28)
Equations (2.26), (2.27) and (2.28) are called the Debye equations, where x = C/G and is
called the relaxation time, such that 1/x = cor, the relaxation frequency. The form of
equations (27) and (28) are shown in figure 2.2 (b). The general dependence of A^on
frequency, emphasizing the various polarization mechanisms, is illustrated in figure 2.3
(Iskander, 1992; Moulson and Herbert, 1990; and Reitz etal., 1980).
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20
- 1
log go
K
logo)
Figure 2.2. (a) Behavior of dielectric parameters near resonance frequency, (b) Behavior
of dielectric parameters showing Debye relaxation.
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21
Space charge
K'
Dipolar
Ionic
Electronic
-
1016
Frequency (Hz)
Figure 2.3. Frequency dependence o f various polarization mechanisms.
2.2 Theory of Barium Titanate
This section describes the crystal structure and symmetry o f BaTiOj, and their
relationship to lattice and thermodynamic theories which attempt to explain the development
of ferroelectricity. Next, the connection between the development of spontaneous
polarization and mechanical strain is established. Then, the static and dynamic properties
of domains are presented, followed by a discussion of the equilibrium properties of
ferroelectric BaTiO,. Finally the effects of shrinking the crystallite size under different
boundary conditions are examined through considering particle and grain size effects.
2.2.1 Crystal Structure and Symmetry
Barium titanate (BaTiO,) undergoes a reconstructive phase transition from a hexagonal
crystal structure at 1460 °C, into the perovskite crystal structure. The perovskite structure
is described by a simple cubic unit cell with barium cations (Ba+2) located at the comers, a
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22
titanium cation (Ti44) located at the body center, and oxygen anions (O-2) located at the face
centered positions. In this configuration, the Ti4 4 cations are surrounded by six oxygen
ions in an octahedral arrangement. The comer positions occupied by Ba4 2 are termed “A—
sites,” and the center positions, occupied by Ti44, inside the oxygen octahedral
configurations are termed “fi—sites.” Alternatively, the structure may also be viewed as a
network of comer-linked oxygen octahedra, with the titanium cations filling the octahedral
holes (T i0 6 groups), and the large barium cations filling the dodecahedral holes. Figure
2.4 emphasizes these alternate views of the perovskite structure. As temperature is cooled
further, the structure undergoes a first order displacive transition to a tetragonal phase at ~
130 °C. The cubic unit cell o f the perovskite structure elongates along an edge (c-axis)
with an associated slight contraction along the other two edges (a-axes). At approximately
0 °C, the tetragonal structure elongates along a face diagonal, transforming into the
orthorhombic structure. At -90 °C, the orthorhombic structure elongates along the body
diagonal, transforming into the rhombohedral structure. Figure 2.5 illustrates the various
unit cell distortions manifested by each polymorphic transition (Jaffe etal., 1971).
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Ba+2
O2
T i+4
Figure 2.4. Cubic perovskite structure.
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24
Cubic
Tetragonal
a \ = a-i - < 2 3
= 4.009 A
d \ —ci2 ^ < 2 3
@ ~ 130 °C
a \ = a2 = 4.003 A
a 3 = 4.022 A
"elongation along as"
Orthorhombic
"pseudomonoclinic"
Rhombohedral
v
\
\
a l - a3 * a2
@ ~0°C
a l = a3 = 4.012
a 1 —a 2 = a 3
@ ~ - 90 °C
a \ = <32 = a 3 = 3.998
A
a2 = 3.989 A; p = 89° 51.6'
"elongation along face diagonal'
A
a =89° 52.5'
"elongation along body diagonal"
Figure 2.5. Unit cell distortions of BaTi0 3 polymorphs.
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25
The point group o f a crystal is the group o f macroscopic symmetry elements that its
structure possesses. Barium titanate, in its cubic configuration, belongs to the m3m point
group or crystal class. The three orthogonal mirror planes make the point group centric;
that is, it possesses a center of inversion symmetry. This symmetry operation moves each
point (x, y, z) to the position (-x, -y, -z). In its tetragonal modification, it belongs to the
4mm crystal class, and this center of symmetry is lifted. A fundamental postulate o f crystal
physics is Neumann’s Principle, which states that the symmetry elements o f any physical
property of a crystal must include the symmetry elements of the point group.
Consequently, the physical properties o f cubic B aT i0 3 must possess this symmetry element
(Nye, 1985).
The location of the Ti* 4 ions in the well shielded and symmetrical oxygen octahedra
appears to be a prerequisite for the ferroelectric state. In its cubic configuration, the
equilibrium position o f the titanium ion is in the center of the oxygen octahedral. The Ti-O
bond is intermediate between ionic and covalent, thus, when the Ti* 4 is displaced by a small
amount, either through an applied field or thermal fluctuations, the resultant dipole moment
has a more than linear increase with distance. Near the cubic to tetragonal transition
temperature, the Ti* 4 can be viewed as moving in a shallow potential well and any
displacement from the center of this well results in large dipole moments. The transition,
therefore, consists of a systematic displacement of the equilibrium position of the titanium
ion from the center o f the octahedron towards the oxygen ion. This results in a net
permanent moment. The octahedra and their moments are coupled by common oxygen
ions, and any displacement of a Ti* 4 towards a specific oxygen ion unbalances the
neighboring titanium ions. This coupling through neighboring ions leads to a feedback
effect until the oscillations become anharmonic. At the cubic to tetragonal transition
temperature, the vibrations are brought into ordered phase relations against any random
thermal vibrations, and the equilibrium position of the Ti* 4 ion shifts. The structure is now
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26
tetragonal and ferroelectric. The development of ferroelectricity is therefore a cooperative
phenomenon, originating from vibrational states, and not from rotating dipoles (Hippel,
1952).
2.2.2 Ferroelectricity
Ferroelectricity is defined as the spontaneous formation and ordered alignment, through
their mutual interaction, o f electric dipole moments which can be reoriented along two or
more different crystallographic directions through the application of a suitable external
electric field. A ferroelectric can also be defined as a material which exhibits hysteresis in
its P versus E-field curves. The occurrence o f ferroelectricity is associated with the
presence of permanent electric dipoles, the phenomenon being somewhat akin to the
spontaneous alignment of rotating permanent magnetic dipoles treated by the Langevin
theory (Langevin, 1905). In an external electric field, permanent dipoles experience a
torque which tends to align them; however, the external fields required to overcome
randomizing effects due to thermal energy are nearly unobtainable. For the electric case, x
is defined as the ratio of field energy to thermal energy and is given by
x=
(2.29)
where k is Boltzmann’s constant and T is temperature. At low fields, the ratio of average
dipole moment to true dipole moment, per molecule, p jp , plotted against x, is nearly
linear, and is approximated by pa I p = ^ x ; therefore,
2
Pa= — E \
a 3kT
(2.30)
where pa is proportional to £ '. If E ' is replaced with Eapp, it can be shown that the field
necessary for dipole realignment is nearly impossible. However in condensed phases, the
field acting on a molecule (local field) is dramatically modified by the surrounding
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27
polarization. The effect is considered by drawing an imaginary sphere around a center
reference molecule, as illustrated in figure 2.6. By treating the surrounding material as a
continuum, it can be shown that the free dipole ends which line the wall of the sphere
contribute a term P I 3e 0 to the orienting field at its center. By neglecting any other
contribution, the Mosotti expression for the locally acting field is arrived at:
J£
(2.31)
0
The externally applied field induces polarization, which in turn increases the local field,
however, the local field then further acts to increase the polarization, which if left
unchecked, would lead to catastrophic polarization. From equation (2.5), polarization,
P = npa, and through (2.30) with E ' — Eloc, given by (2.31),
p2
P
f>= 'lJ± - ( E aD„+ ----->•
3IcT app 3e„
(2-32)
1
i
i
i
i
+
+ X
1
*
1
▼
1
f
1
T
V
1
V
i
i
\ //'
+
\\
▼ W
i
i A
▼ f
* //
i
i I\\ - E.loc = E* +---"
T ▼
*
+
i
i
+ i
I
V
app
1
T ▼
i
i
T T
i
▼ +
I
1
I
I
Figure 2.6. Model for the calculation of the locally acting internal field.
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28
By rearranging (2.32) and letting E'
P =~ ~ %
= E,
E
(2-33)
In equation (2.33), Tc = np219k£0. Equation (2.33) illustrates how the randomizing
effects due to thermal energy keep the polarization under control until some critical
temperature is reached, Tc, at which point the randomizing effects of temperature are
overcome, resulting in spontaneous polarization. From (2.6) and (2.33),
P
3T
=X =K-1 =
e0E
T -T c
(2.34)
At Tc, called the Curie temperature, the electric susceptibility approaches infinity. The
linear dependence of \!%e on (T-Tc) for T > Tc is known as the Curie-W eiss law (Hippel,
1950, 1952).
An interpretation of ferroelectricity can also be accomplished through a thermodynamical
model (Devonshire, 1949, 1951;and Fatuzzo and Merz, 1967). In thisapproach, the
independent variables are electric field, E, stress, X, and temperature, T; the dependent
variables are polarization, P, strain, jc, and entropy, S. The differential of internal energy,
dU, of a polarizable deformable solid insulator, subject to an external stress and electric
field is
dU = TdS + Xdx + E dP .
(2.35)
The elastic Gibbs potential function is defined as
G, = U - T S - X x.
(2.36)
From (2.35) and (2.36),
dGx = EdP - SdT - xdX.
(2.37)
Under constant temperature and constant stress, dG x = EdP, thus, E = dGx/d P . The
elastic Gibbs function can now be expanded in its most general expression, in powers of
polarization:
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29
Gx = a.iPi + ^ a ijPiPj + ^a .,jkPiPj Pk + ^ a ijilPiPj PkPl + - .
(2.38)
The CLiJk... terms are tensor coefficients of rank given by the number of suffices; they
transform as products o f vectors and are assumed temperature dependent. The ferroelectric
is a polar subgroup o f the parent (paraelectrie) prototype. The thermodynamic approach
looks for a polar state in equilibrium with the non—polar state. Start by assuming the
electric field is directed along a certain axis, such that E and P are co-linear, and also that
the material has a center of inversion symmetry (characteristic of the centro-symmetric
m3m point group), i.e., ± fields must give rise to equal ± polarizations, such that all odd
power terms must be zero, equation (2.38) becomes,
G\ = J a np + ^'a ini^> + g a i u i n ^ •
The dielectric stiffness,
T|
=
=
T |,
(2.39)
defined as the inverse susceptibility (1/%), is given by (2.40),
ocn +3cci m / > + 5 a m m / > .
(2.40)
In the non—polar state, P = 0 when E= 0, therefore in the non-polar state, q —a , , . Now
assume that Curie-W eiss behavior is obeyed in the non-polar state, such that,
q = q 0 ( T - r 0).
(2.41)
In equation (2.41), the value T0 has been employed, where the temperature T0 is close to,
but not necessarily equal to the Curie temperature. The constant T0 will be called the Curie
point. Equation (2.39) can now be rewritten as
c , = l t l o ( r - r 0 ) P 2 + 4 p / >4 + 7 Y />6,
2
4
6
(2.42)
with
t)G
E = - ^ - = x\q( T - T q)P + PP3 + yP5,
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(2.43)
30
where a , ,,, and a , ,,,,, have been replaced with P and y, respectively. The highest order
term is assumed positive, otherwise catastrophic polarization would result in the lowering
o f the free energy. Again, in the non-polar state (P = 0); from (2.42) G, = 0. If there is a
polar state, characterized by spontaneous polarization Ps, in equilibrium with the non-polar
state at some temperature Tc, it follows that,
O ,
= o = i i i 0( r t - r 0)/>,! + i p / ^ + i Y^ .
2
4
6
(2.44)
It must also be, that if the polarization is spontaneous, it persists even in the absence o f an
electric field, therefore,
* i = o = ri0(
£ ' = ;~
r
,
+
y
p ; .
(2.45)
By solving (2.44) and (2.45) simultaneously, it can be shown that,
Ps = t ( ~ “ )»
4
y
(2-46)
and that at Tc there is a discontinuous step in polarization such that,
3B2
t1o(^c-7’o) = T7 - 16y
(2-47)
Equation (2.47) was obtained by inserting the form of P f given by (2.46), into (2.44).
The discontinuous step just described is indicative of a first-order phase transition.
By setting (2.43) equal to zero and solving for P, the spontaneous polarization is given by,
+ {J -4 v
n
0p
“2( r -
) } '' 2 ] } '' 2 -
(2.48)
Finally, from (2.42) the inverse susceptibility is obtained as
Tl = ^
= Tl0 ( r - r 0) + 3pP 2 + 5yP5.
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(2.49)
31
Above the Curie temperature and in the absence of large applied fields, the polarization is
zero, and (2.49) describes Curie—Weiss behavior. Below the Curie temperature, P can no
longer be neglected and is set equal to Ps given by (2.48).
Figure 2.7 shows the shape o f the G, versus P curves for various temperatures. Curve
a depicts G, vs P at some high temperature, T »
Tc. This curve exhibits a single
minimum, corresponding to P = 0, characteristic of the paraelectric phase. Curve b is for a
temperature above Tc, yet very close to Tc. The deep minimum corresponding to P = 0 is
still present, however, two outer inflection points are visible. The non-polar phase is still
stable at this temperature, however, the ferroelectric phase may be induced by an applied
field. Curve c corresponds to T = Tc. Here, three minima o f equivalent depth are present.
The central minimum, as previously discussed, corresponds to the non-polar phase. The
outer minima are due to the ferroelectric phase and correspond to two opposite values of the
spontaneous polarization, +PS and -Ps, each of which yield the same free energy. At this
temperature, the non-polar and ferroelectric phases are in equilibrium. Curve d
corresponds to T0> T > Tc. Here, the outer minima corresponding to the ferroelectric
phase are deeper than the central minimum corresponding to the non-polar phase,
indicating that the non-polar phase is only metastable at this temperature. Finally, curve e
depicts the case for T < T0. All that remains are the two deep outer minima corresponding
to the ferroelectric phase, characterized by two opposite values o f spontaneous polarization.
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32
Figure 2.7. Free energy functions at various temperatures for a ferroelectric undergoing
a first-order transition.
Figure 2.8 illustrates the behavior o f spontaneous polarization, dielectric constant, and
inverse susceptibility, for a first-order ferroelectric, through the paraelectric to ferroelectric
phase transition. Upon cooling from the paraelectric phase, r\ follows the Curie-Weiss
law, decreasing linearly with decreasing temperature; the temperature at which the
extrapolated rj intersects the temperature axis is T0. Dielectric constant, K, increases, and P
= 0. At r c, K traverses a sharp, finite maximum, and P exhibits a discontinuous increase
to Ps. On heating, the temperature at which P discontinuously drops to zero is Tc.
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33
/
>»
I
I
Temperature
Figure 2.8. Properties as a function of temperature for a first-order ferroelectric.
All crystals possess fundamental elastic vibrations. The normal modes of these
vibrations include the higher frequency transverse optical (TO) and longitudinal optical
(LO) branches, as well as the lower frequency longitudinal acoustic (LA) and transverse
acoustic (TA) branches. All vibrations in a crystal can be described by linear combinations
of these fundamental modes. A lattice dynamical model (Cochran, 1960, 1961, and
Fatuzzo and Merz, 1967), based on the theory that ferroelectric transitions are due to an
instability in a particular normal lattice vibrational mode, was developed, and well describes
the ferroelectric transition in B aT i0 3. The theory makes use of the “core-shell”
representation of atoms and ions. Considering a cubic lattice, equations of motion for the
cores and shells, taking into account short-range core-core, core-shell, and shell-shell
interactions, as well as long-range core-core, core-shell, and shell-shell interactions, are
established. By neglecting the mass of the shell, it was shown that for time varying
oscillations, solutions corresponding to oof =
0
and cOj = ^ were possible, where co is the
angular frequency o f vibration. The solution cof = 0 represents the intersection of the
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acoustic branch with q = 0, q being the wavenumber, where q = 2 n /X . The solution
co2 —0 represents a degeneration o f one o f the optical branches when q approaches zero.
This degeneration indicates an instability o f one particular mode o f vibration, and can be
connected with the occurrence of ferroelectricity. Cochran showed that atq = 0, for a cubic
lattice of n atoms per unit cell,
(2.50)
where Ks is the static dielectric constant, Ke is the square of the refractive index, and (a>y)L
and (o)y)r , are the frequencies of the longitudinal and transverse optical branches,
respectively. Equation (2.50) says that if one of the transverse optical branches vanishes as
q —>0, at a certain temperature, Ks approaches infinity.
A situation o f this type is approached for the optical mode o f lowest frequency in some
cubic structures. In a wholly or partly ionic cubic crystal, lattice vibrations result in
polarization oscillations. Consequently, the polarization oscillations result in fluctuations in
the local field strength. This local field interacts with ions through long-range Coulomb
forces. If these long-range Coulomb forces act against the short-range restoring forces,
and if the magnitude of these two competing forces becomes nearly equal, they will tend to
cancel one another out, resulting in a very low net restoring force. This unusually low
restoring force results in an unusually low frequency for this mode, much lower than
would be expected for an optical mode. It can be shown that in the harmonic
approximation the crystal would be unstable against these low frequency vibrations, and
undergo a transition to another structure. This represents the paraelectric to ferroelectric
(cubic to tetragonal) transition which acts to raise the frequency o f the so called ferroelectric
soft-mode. The frequency o f this mode, determined from infrared absorption experiments,
is termed the critical frequency, v F, and has a wavelength of - 296 p.m in BaTi03. The
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35
critical frequency was shown to be temperature dependent, and at the temperature where
v F reached a minimum (Tc), Ks peaks to a maximum. The temperature dependence of v F
is given as follows:
v2
f oc( T
- T c ).
(2.51)
2.2.3 Electromechanical Coupling
It has been shown that ferroelectricity is characterized by the development of
spontaneous polarization. Associated with this spontaneous polarization is a lattice
distortion which leads to strain. The coupling o f this strain to applied electric fields or
changes in polarization is next discussed.
Stress is defined as an applied force per unit area. Consider a unit cube within a body
with its edges parallel to the orthogonal axes x v x 2, and jc 3. A force is transmitted across
each face o f the unit cube by the surrounding material. This force across each face may be
resolved into three components. The symbol <r,y denotes the stress arising from the
component o f force in the x, direction transmitted across that face of the unit cube which is
perpendicular to x}. Stress, <j,y, is thus a second-rank tensor which expands into a 3 X 3
matrix. The diagonal terms, i = j , are the normal components, and the off-diagonal terms,
i * j , are the shear components, where a,y = c y, for i * j.
Strain is defined as the rate o f change of displacement with distance, etJ = d u j d x j , and
is a dimensionless quantity. In general, a strain can be described as a true deformation plus
rigid—body rotation through some small angle. In the absence of true deformation, a rigid—
body rotation should not result in strain; i.e., the ei} tensor should vanish. This is ensured
by expressing etj as the sum of a symmetrical and an antisymmetrical tensor, where
eij = e ,> +
• The symmetrical part is E(> =
(e,y + ey,) and the unsymmetrical part is
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36
etj -
Choosing the symmetrical part of the sum to define strain ensures that
rigid-body motion results in no strain. The strain tensor, £,y, is represented as thus,
£j,
£ ,2
£ 13
£ 12
£ 2 2
^23
_ £ 3i
£23
£33
(2.52)
The diagonal components are the tensile strains, while the off-diagonal terms indicate shear
strains.
If a stress is applied to a ferroelectric material having spontaneous polarization, a change
in spontaneous polarization in proportion to the applied stress may be observed. This is
called the direct piezoelectric effect. If P represents some change in spontaneous
polarization, piezoelectricity is described by P = do, where d is called the piezoelectric
modulus. Explicitly,
(2.53)
where
is a third—rank tensor. The djJk tensor has 27 coefficients, however, due to the
fact that dijk is symmetrical in j and k, 9 o f the 27 coefficients are eliminated as independent
components, leaving 18 independent dijt. Consequently, a more concise notation known as
matrix notation is employed. The first suffix o f the original notation, /, is left unchanged,
but the second and third suffixes (t and j ) are replaced by a single suffix running from
6
1
to
, where 11 —» 1, 22 —» 2, 33 —» 3, 23, 32 —» 4, 31, 13 —> 5, and 12, 21 —>6. A sim ilar
change is made in the notation for the stress components. In the new notation,
= dip j
(i = 1, 2, 3; j = 1 , 2, 3, 4, 5, 6 ).
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(2.54)
37
W hen an electric field is applied across a piezoelectric crystal the shape o f the crystal
changes slightly. This is called the converse piezoelectric effect, and relates strain to an
applied electric field. In matrix notation the effect is described as,
(2.55)
where the du coefficients connecting the field and the strain in the converse effect are the
same as those connecting the stress and the polarization in the direct effect.
At very high strains, non-linear effects must be considered, and are accounted for by
introducing additional quadratic terms into (2.55). Thus, at high strains, the converse
effect, in full tensor notation, may be written as,
£,y —diJkEk + M ijlk
jJlkEkE[
k [’
(2.56)
where M ijlk is the electrostrictive coefficient and is a fourth—rank tensor symmetrical in / and
j, and in I and k. In the theoretical treatment of the solid state, again by expansion of the
elastic Gibbs free energy function (2.38), it can be shown that strain is expandable in terms
of polarization, where
(2.57)
In this case but and Q'jkl are the piezoelectric and electrostrictive coefficients, respectively
(Nye, 1985; and Devonshire, 1949).
Finally, when a solid is subjected to a stress, a small deformation is introduced,
resulting in strain. If H ooke’s Law is obeyed, the strain is proportional to the stress, such
that e = j<y, where j is a constant called the elastic compliance. Alternatively, a = ce,
where c is called the elastic stiffness. The generalized form of Hooke’s Law in full tensor
form is as follows:
£|) —SijkPkl»
or alternatively,
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(2.58)
38
<*,/ = cijk£t,-
(2.59)
The elastic constants, compliance and stiffness, are fourth-rank tensors and are each
represented by 81 constants. Due to the symmetry in the first two and last two suffixes,
the number o f independent components is 36, and the matrix notation may again be
employed. It follows then, that
e, = SjjGj
(/, j = 1 , 2 , ..., 6 )
(2.60)
(i j =
(2.61)
and
ct, = CjjEj
1
, 2 ...... 6 ).
Because this work deals mainly with B aT i0 3 in its cubic or tetragonal modifications,
important relations between
and ctj for these two systems are given below:
Tetragonal system:
~
S 3 3 IS 1
C 33 — (^11
^ll~ ^1 2 =
^ 12 ) ^ ’ C 44 =
1/(^11
12 ^ ’ ^ 1 3 = ”^ 1 3 ^
1 ^ 4 4 ’ C66 =
^ J 66»
where s = j 33( 5 , , + sl2) - 25,23.
(2.62)
Cubic system:
^
|+
*^12
(•*11 — ■Sl 2 ) ( f U + 2 J 12)
C = ______ __________
( 5 I1
+ 2 sI2)
C4 4 = 1 /^4 4 .
(2.63)
2.2.4 Domains
When BaTi0 3 cools through its Curie temperature, the initially cubic structure provides
six possible directions, corresponding to the three pairs of antiparallel directions along each
of the cube edges, along which the polar tetragonal axis can develop with equal probability.
This results in the formation o f domain or twinning patterns. Domains are three
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39
dimensional regions within a ferroelectric crystal which contain uniformly aligned
permanent dipoles. The domains are separated by domain walls, which are defined as the
loci o f the points where the dipole orientation suddenly changes. The tetragonal phase of
B aT i0 3 gives rise to two different types of domain walls: those separating antiparallel
dipoles (180° walls) and those separating dipoles at right angles to each other (90° walls).
Under equilibrium conditions (zero stress and electric field), domains arrange
themselves in such a way that the divergence of polarization (given by (2.17)) equals zero,
V • P = 0, in the bulk o f the crystal. Because of the spontaneous deformation associated
with the development of spontaneous polarization, it is possible that adjacent domain pairs
(distinguished by different orientations of Ps) give rise to nonuniform strains. The
conditions predicting the preferred orientation of domain walls were established using
mechanical compatibility and polarization wall charge criteria (Fousek and Janovec, 1969).
The mechanical compatibility condition requires that the spontaneous deformation in each
domain, defined by one of two conjugate Ps vector pairs, must be equal at the plane
separating the domains. Conjugate Ps vectors are defined as two Ps vectors lying along two
crystallographically equivalent directions in the paraelectric phase. The strains, meld, are
governed by the piezoelectric and electrostriction coefficients as follows:
mek l= d iklmPl +QljklmPimPj,
C2-64)
where m = 1,2, corresponding to each Ps vector. An arbitrary infinitesimal vector, d s , in
the paraelectric phase, changes length with the onset o f the ferroelectric phase. The
difference between the square o f its length in the paraelectric and ferroelectric phases for
each Ps vector is established using (2.64). By equating the difference obtained for each Ps,
and requiring that ds must lie in the (hlcl) plane of the domain wall, a differential strain
tensor, A u , is arrived at. From a given prototype paraelectric phase and two conjugate Ps
vectors, all At/s are found. The strain condition for B aT i0 3 (m3m prototype) establishes
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40
no preferred domain wall orientation, i.e., the domain wall separating conjugate Ps vectors
can have arbitrary orientation. Because of this arbitrary orientation, the wall may either be
charged or neutral. Since an uncharged wall is energetically more favorable, it is likely that
walls will orient so as to maintain charge neutrality. Figure 2.9 depicts equilibrium domain
orientations through a thickness of a ferroelectric crystal, with its opposing surfaces
electroded, possessing 180° and 90° domains.
180 domain walls
9 0
° domain walls
Figure 2.9. Static walls separating 180° and 90° domains.
If an electric field is oriented opposite to the domain orientation, the polarization will be
reversed. Polarization reversal involves several mechanisms: 1) nucleation o f new
domains; 2) forward growth of domains; 3) lateral expansion of domains; and 6 )
coalescence of domains. If a voltage pulse is applied across a ferroelectric crystal, over
some period of time, the polarization will reverse, giving rise to a polarization current (see
(2.7)). The polarization current increases to a maximum, 1 ^ , and then decreases to zero.
It has been shown (Merz, 1956) that for electric fields between 1 to 15 kV/cm,
W = 'o exp(-oc / E)
(2.65)
and the switching time, ts, is given by
ts = r 0 e x p (a /£ ’),
( 2 .6 6 )
where iQand t0 are constants, and a is the activation field. At higher fields, between 10 to
100 kV/cm, the switching time was observed to follow a power law of the type
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41
= kE ~n,
(2.67)
where £ is a constant and n ~ 1.5; the very small switching times measured under the
highest fields, revealed domain walls can move at supersonic speeds (Stadler, 1958). The
dependence of domain wall velocity, v, has been shown to vary with electric field. A t low
fields (E < 1 kV/cm),
v = v„ exp(—S / E),
( 2 .6 8 )
where v„ and 5 are nearly constant, showing slight variations with E (Miller and Savage,
1959). At high fields (E > 1 kV/cm),
v = k E 14,
(2.69)
very nearly equal to the inverse switching time given by (2.67) (Stadler and Zachmanidis,
1963).
2.2.5 Properties of Ferroelectrics
A description of the phase transformations in BaTiOj was presented in 2.2.1. The
corresponding changes in the spontaneous polarization and the dielectric constant are
shown in figure 2.10. In figure 2.10 (b), note that the reported dielectric constants below
the rhombohedral phase transition are inconsistent with the symmetry of that phase.
j
i _________
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42
(a)
e
D
<c
0.25
a
o
0.20
u.
_2
0.15
o
a.
r.
3
o
4J
S
0.10
0.05
-
1 4 0 - 1 2 0 -1 0 0 - 8 0
-6 0
-4 0
-2 0
0
40
60
80
100 !
T<
Temperature °C
(b)
©
X
e
c
o
U
u
b*
C
<uJ
u
s
a axis
axis
-1 6 0
-1 2 0
-8 0
-4 0
120
Temperature °C
Figure 2.10. (a) Spontaneous polarization versus temperature; and (b) dielectric constant
measured along the a and c directions versus temperature (Merz, 1949, 1953).
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Figure 2.11 shows the general shape of a P vs E hysteresis loop for a ceramic ferroelectric.
Where the curve intersects the P-axis at E = 0 is the remanent polarization, P r. The
spontaneous polarization is obtained by extrapolating the polarization at the highest fields
(termed saturation polarization, P J) back to zero field, along the tangent, and is essentially
equal to Pr for single crystals. The coercive field strength, Ec, is obtained where the loop
intersects the E-axis at P = 0, and is defined as the field strength necessary to switch the
domain state. Its meaning is somewhat ambiguous because, as was shown by (2.66),
there is no threshold field required for switching. Switching will occur for any applied
field, no matter how small, provided sufficient time is allowed. In other words, at fields
less than E . switching still occurs, but over longer time periods.
P
Figure 2.11. Dielectric hysteresis loop.
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44
2.2.6 Size Effects
It has been shown that ferroelectric B aT i0 3 crystals undergo domain twinning upon
cooling through the paraelectric-tetragonal transition, consequently, B aTi0 3 may exhibit
complex domain structures, possessing several different types o f domains (180° and 90°).
This section examines what happens to the domain structure when first the free crystallite
(particle) size is decreased and then when the mechanically clamped crystallite (grain) is
decreased.
2.2.6.1 Particle Size Effects
As particle size is decreased, the volume energy of the particle decreases. At very fine
particle sizes, the surface energy term becomes increasingly important. At large particle
sizes, the total energy is reduced by twinning, where the volume energy term is decreased
at the expense o f interfacial domain wall energy, the sum of which is less than the
untwinned particle. At some size, it must be that, domain wall interface energy becomes
irrecoverable from the volume energy term, below which twinning cannot lower the total
energy. This, of course, should result in some rather interesting size dependent effects on
the domain state, the study of which is called size effects.
Ferroic crystals have movable domain walls which result in hysteretic behavior.
Depending on the driving field necessary to move the domain walls, the primary ferroics
are classified as ferromagnetic, ferroelectric, or ferroelastic. Twinning in these materials,
therefore, reduces the energy of destabilization fields, whether they be magnetic, electric,
or elastic. In terms of size effects, ferromagnetic materials are by far the best studied, and,
it is anticipated that the other ferroic classes will follow the magnetic analog as size is
reduced. That being the case, several transformations are expected with reducing size in
ferroelec tries.
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45
At large particle sizes, as discussed above, twinning lowers the volume energy o f the
particle, with the appearance of complex domain structures possessing several types of
walls. As size is reduced further, it becomes increasingly difficult to recover the wall
energy from the volume term, consequently, it is expected that the number of domains will
decrease, at first by one type, and then the others. The first transformation is then from a
polydomain particle to a single domain particle. Upon further reduction in size,
ferromagnetic particles undergo a transformation to the high temperature symmetry group
exhibiting enhanced responsiveness. This state is termed superparamagnetic, and is
characterized by a zero net magnetization, the disappearance o f a magnetic hysteresis loop,
and extremely high magnetic susceptibilities. The electric analog to superparamagnetism
may be found in the family o f relaxor ferroelectrics. Compositions including many o f the
A fB ^ B ^ lO ja n d A fB ^ B '^ O j perovskites exhibit microdomains (~ 20-300
A) o f 1:1
ordering on the B sublattice dispersed in a disordered matrix. As a result o f this
nonstoichiometry, it has been suggested that the spontaneous polarization in these materials
is also disordered on a very fine scale, thus the ceramic may be regarded as a collection of
disordered, but highly orientable, dipoles. The result is a phase possessing a high
dielectric permittivity over a broad temperature range, with zero net spontaneous
polarization. This superparaelectric phase would then be expected to behave as an
unpolarized, but highly orientable single domain, possessing a high dielectric constant.
Again, returning to the relaxor system, a critical domain size at which antiferroelectric
ordering appears has been identified in lead indium niobate to be 80 nm. This size may
also serve as a guideline for the dimension at which ferroelectric particles show
superparaelectric behavior. Direct observations o f superparaelectric behavior have not been
made. Finally, since ferroelectricity is a cooperative phenomenon, it is reasonable to
suppose that the system will be forced to revert to a paraelectric state at some size at which
there are simply too few atoms to sustain the cooperative interactions necessary for
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46
ferroelectric behavior. In summary, the size dependence of ferroelectries should exhibit
four states, predicted from the magnetic analog: multidomain, single domain,
superparaelectric, and paraelectric (Newnham and Troiler-M cKinstry, 1989).
Various workers determined the particle size at which the room temperature tetragonality
(c/a lattice constant ratio) of B aT i0 3 approached unity. In this sense, critical size, dml,
refers to the particle size where the room temperature tetragonality equals
1
(cubic) or
equivalently, the cubic to tetragonal transition temperature is lowered below room
temperature. Reported values of
are summarized in Table 2.2.
Table 2.2 Measured Values of J crit
Reference
(nm)
15
1 0
<70
1 2 0
25
Anliker et al., 1954
Tanaka etal., 1962
Muller, 1990
Uchino et al., 1989
Criado et al., 1990
Uchino (Uchino et al., 1990) showed that decreasing particle size led to a systematic
reduction in tetragonality, with room temperature tetragonalilty decreasing to unity (cubic)
at a dml = 0.120 p.m (figure 2.12). The temperature dependence of the d a ratio showed that
with increasing temperature, the tetragonality decreased, and exhibited an abrupt decrease
into the cubic state, thus, first order transition behavior was retained (figure 2.13). In
addition, the Tc of sintered bulk samples of B aTi03, having various grain sizes were
compared to the Tc o f various B aT i0 3 particles of different sizes. The sintered samples, in
general, showed a lower Tc than that o f the powders o f the same particle size. Both
ceramic and free particles, showed a decrease in Tc with size. The decrease in Tc was
attributed to hydrostatic compressive stresses generated from sintering and surface tension
(ceramic case) or from surface tension (free particle). A quantitative model relating
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“effective” surface tension, y, to internal pressure, P, induced by the surface tension was
proposed,
P = 2?//?,
(2.70)
where R is the particle radius. Surface tension, 7 , was estimated to be 50 N/m for a
0.1 pm. For nonpolar oxides (e.g. MgO),
7
— 1 N/m. The large difference in 7 estimated
here for fine particle BaTiO, suggests that the surface tension of an electrically polarized
crystal may be very different from that o f a nonpolar crystal.
Single crystal
1.010
1.008
1.006
& 1.002
1.000
0.2
0.3
0.4
=
0.5
0.6
0.7
0.8
0.9
Particle s iz e ( u m)
Figure 2.12. Room temperature tetragonality change with particle size in B aT i0 3 powder
(Uchino etal., 1990).
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48
Particle size ( p m )
002
0.04
0.06
0.08
0.6
0.8
0.10
500
400
Pow der
S i n t e r e d bulk
S 100
0.2
0.4
P a r t i c l e / Grain s i z e ( p m )
Figure 2.13. Particle and grain size dependence of the Curie temperature in B aT i0 3
(Uchino et al., 1990).
2.2.6.2 Grain Size Effects
Intuitively, grain size effects are expected to differ from particle size effects mainly
because the boundary conditions which govern a grain assemblage differ from those which
govern a free particle. Most significantly, grains in a ceramic are three dimensionally
clamped. In the case of BaTiO,, as the ceramic cools through the cubic-tetragonal phase
transition, a tetragonal lattice distortion is introduced, resulting in large strains in the
individual grains. This leads to rather complicated domain structures. Typically, large
grain (> 20 (i.m) polycrystailine ceramic BaTiOj exhibits a room temperature dielectric
constant, Ka ~ 1500-2000. Various investigators, however, have observed Kn to increase
with decreasing grain size (Kniekamp and Heywang, 1954; Jonker and Noorlander, 1962;
and Sharma and McCartney, 1974), peak at some critical size, and then decrease with
continued size reduction (Brandmayr et al., 1965; Kinoshita and Yamaji, 1976; Arlt et al.,
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49
1985: and Arlt, 1990,1990). Figure 2.14 shows the temperature dependence of dielectric
constant in various grain size high purity BaTiO, ceramics (Kinoshita and Yamaji, 1976).
Clearly Kn increases from ~ 1500 for a grain size of 53 pm to > 5000 measured for a grain
size of 1.1 pm . The lower tetragonal-orthorhombic, and orthorhombic-rhombohedral
transition temperatures appear to increase slightly with decreasing grain size. The cubictetragonal transition temperature and paraelectric behavior seem grain size independent
through the investigated size range.
x
£Z
3=55
c
©
U
u
-150
-100
-50
0
50
too
150
Temperature (°C)
Figure 2.14. Tem perature dependence of dielectric constant in various grain size, high
purity BaTiO, ceramics (Kinoshita and Yamaji, 1976).
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50
The anomalous increase in Kn with decreasing grain size has been attributed to both an
increase in domain wall density with decreasing grain size (down to ~
1
pm ) as well as an
increase in residual internal stress (for submicron grain sizes). Regular twinning through
the formation of 90° domains below the cubic-tetragonal structural phase transition is the
means by which internal stress energy is minimized. Several workers have attempted to
correlate the 90° wall density with grain size. It has been proposed that the width of 90°
domains is fixed at 1 pm , which suggests that no 90° domain wall can exist in submicron
B aT i0 3 ceramics (Buessem et al., 1966). Consequently, this would result in unrelieved
high residual internal stress in submicron B aT i0 3 ceramics. Using a modified Devonshire
thermodynamic treatment, Buessem showed that this high internal stress should result in an
increase in Kn, in accordance with observations.
A quantitative model has shown the twinning associated with stress relieve to be rather
complicated (Arlt, et al., 1985; and Arlt, 1990, 1990). For grain sizes greater than ~ 10
pm, twinning through the formation of both 180° and 90° walls accounts for a three
dimensional correction, such that a tetragonal lattice distortion is completely accommodated
within a cubic grain. Domain width, d, is expected to be proportional to (g)m, where g is
grain size. This three dimensional adjustment o f the grain accounts for the well known
“herring bone” domain patterns exhibited in large grain BaTi0 3 ceramics.
At grain sizes less than ~ 10 pm, 90° twinning allows only for a two dimensional
correction; i.e., the cubic grain retains its gross shape in two dimensions (square) by
twinning; the third dimension is left uncompensated. An additional combination of
longitudinal stresses is required in order that the tetragonal lattice distortion is
accommodated in a cubic grain. The general formalism is based on the total energy
minimization, where
wtol = wM+ h'£ + w b,+ w'j = minimum.
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(2.71)
51
The elastic energy density, the electric energy density, the domain wall energy density, and
the surface energy density are given by wM, wE, ww, and ws, respectively. In the model,
the surface energy density was considered only to be important at very small grain sizes
and the electric energy density was considered negligible. At the cubic to tetragonal
transition, a tetragonal lattice distortion is introduced with the development of Ps along the c
axis. This results in spontaneous deformation, designated SQ. The strain tensor describing
S 0 is,
fs a
So =
0
0
o'
0
0
If the cubic lattice parameter is a0, and the tetragonal lattice parameters are given by a and c,
Sa = (a - a0)la0 and Sc = (c - aQ)/a0. Consider a cubic grain, o f side g, whose edges
correspond to a coordinate system rotated 45° about the x2 axis. In this rotated system,
spontaneous polarization results in the deformation indicated in figure 2.15 (a). The
deformed grain can be undeformed by one of two ways: ( 1 ) a shearing force plus a
longitudinal adjustment along x2, resulting in a highly strained cube possessing a high
elastic energy; or (2 ) twinning with an equivalent longitudinal adjustment to bring the grain
into a nearly cubic configuration. Twinning o f the grain reduces this high elastic energy, as
exemplified in figure 2.15 (b). In figure 2.15 (b), twinning brings two dimensions o f the
grain into nearly the shape of a square, except for the formation o f the serrated sides. A
combination of compressive (T, = T2) and tensile stress (T2) is required to bring the third
dimension, and subsequently the grain, into a nearly cubic configuration. The total energy
for the configuration in figure 2.15 (b) includes the longitudinal adjustment term, a term
associated with the interfacial energy of the domain walls, and a term associated with the
strains introduced at the serrated sides. The twin width, d, depends on the minimum of the
total energy. The elastic energy due to the serrated sides was show to be,
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52
wSK= 2 k c ffd lg ,
(2 .7 3 )
where (3, is defined in figure 2.15 (b), g is the grain size, and the product kc —c, ,/48. The
total energy density caused by the surface energy of the domain walls is
= o ^ d , where
On is the domain wall energy per unit area. The minimum condition for the sum then
requires that,
d =
(2.74)
y lk c ftj
and
8togo£pf y
V
s
J’
(2.75)
where w2 is the energy density associated with the longitudinal adjustment. Equation
(2.74) exhibits a g in dependence in d.
•H
Figure 2.15. (a) Deformation introduced in cubic grain due to spontaneous polarization;
(b) internal elastic energy minimization through 90° twinning.
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53
Equation (2.74) has been shown to be valid to grain sizes down to ~ 1 (im (figure 2.16).
Below - 1 pm , it is suggested that changes in crystal structure, spontaneous polarization,
and spontaneous deformation, invalidate (2.74). It is however o f interest to extrapolate
(2.74) to g ^ , where gcrj[ = d. By doing this, a hypothetical
= 40 nm was determined.
This would correspond to the size at which the grain becomes monodomain in the
hypothetical case. Finally, using (2.74), and through direct observation of domains in
ceramics, the domain wall energies were determined to be
= 2—5 10" 3 Jm '2.
Finally, at even smaller grain sizes, the energy of the twinned configuration exceeds that
of the untwinned configuration, such that domain wall density decreases, Kn decreases,
and the grains presumably approach a monodomian state. A decreased frequency of 90°
twinning was observed in fine-grained Dy-doped B aTi03; at an average grain size of 1.25
pm, only 10% o f the grains showed 90° twins (Yamaji et al., 1977, 1981). Figure 2.17
illustrates the behavior o f K (relative permittivity, £r) at 25 °C and 75 °C with decreasing
size (Arlt, 1990). In figure 2.17 it can be seen that as grain size is decreased, twin density
increases which is reflected in an enhanced extrinsic contribution to Kn. At some critical
size, additional twinning does not lower elastic energy, below which the twin density
decreases. The domain wall contribution to K n must also decrease. This critical grain size
where K n peaks, gp, occurs at ~ 0.7-1 pm.
Frey (Frey, et al., 1997) considered in detail the role of nonferroelectric interfaces in
ultra-fine grain ceramic B aTi03. In this study, starting powders o f primary crystallite size
approximately equal to 30 nm and free o f any detectable hydroxyl defects were prepared by
an alkoxide decomposition route. Ceramics were prepared from the powders by pressing
at 8 GPa and heat treating at 700 °C, using an multi-anvil press. Subsequent annealing in
air yielded dense ceramics (> than 98 % theoretical) of average grain size between 70 nm
and
2 0
pm.
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54
The grain size dependent Curie-W eiss characteristics and dielectric properties in the
ferroelectric state were measured. In this work, a significant enhancement in the apparent
dielectric constant, K , was observed upon decreasing grain size from 1.7 pm to ~ 0.5 pm,
where K reached a maximum o f ~ 4000; below 0.5 pm , K appeared to decrease,
however, remaining above 2000 for a grain size less than 100 nm. The Curie-W eiss
behavior showed that all grain sizes yielded essentially the same Curie constant, with no
significant change in Tc for grain sizes below 2 pm.
These results were clearly interpreted within the context of the internal stress model
previously discussed (Buessem, et al., 1966), and by treating the ceramic as a diphasic
dielectric comprised of isolated, highly stressed ferroelectric grains surrounded by a
continuously connected nonferroelectric grain boundary region. From high resolution
TEM studies the boundary thickness was approximated to be
8
A, with a dielectric constant
of ~ 130, based on reported values for titania and other nonferroelectric titanates. By
applying a series dielectric mixing law, it was clearly demonstrated that the decrease in K
measured in ceramics with average grain sizes below ~ 0.5 pm was exclusively due to a
series dilution arising from the grain boundary region. W ith decreasing grain size, it was
shown that the dielectric constant o f the ferroelectric grains increased to -4800 at - 0.5 pm,
and remained constant for all grain sizes less than - 0.5 pm down to ~ 70 nm. This work
supports that in the submicron grain size region, domain twinning significantly decreases
with an associated increase in internal residual stress, which acts to enhance K . For grain
sizes less than 0.5 pm, it was proposed that the ceramics were comprised of substantially
single-domain, highly stressed grains. Upon further reduction in average grain size, an
increased dilution effect from the boundary regions was observed. At larger grain sizes
(> 0.5 p), stress relaxation by domain formation led to decreasing values of K .
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55
e
0.2--
. -•
0 10
0.02
0.3
1.0
10
100
g ((im)
Figure 2.16. Domain width of B aT i0 3 as a function of grain size (Arlt, 1990).
7000+
5000- -
o iwosurt d at 25*C
a measured at 70*C
3000
TOO
Figure 2.17. Change in dielectric constant (relative permittivity, £r) with grain size for
BaTiOj ceramics (Arlt, 1990).
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CHAPTER m
MICROWAVE DIELECTRIC MEASUREMENT TECHNIQUES
The purpose of this chapter is to develop several measurement techniques suitable for
the determination of dielectric properties at microwave frequencies. In doing this, this
chapter also introduces the fundamental parameters and terminology used in microwave
engineering together with the basic problem-solving algorithm used in the solution of fields
along transmission lines and waveguides. This generally involves solving M axwell’s
equations under a set of appropriate boundary conditions defined at the guide walls and
material interfaces. Consequently, the initial sections o f this chapter proceed in a “step-bystep” manner while subsequent sections move more rapidly to final solutions, assuming the
reader is already familiar with the basic approach to the solution. By employing Maxwell’s
equations, wave propagation along transmission lines and waveguides is first explored.
These concepts are then extended, and through the application o f appropriate boundary
conditions, lead to the development o f several microwave measurement techniques.
3.1 Microwaves
Microwave refers to alternating current (ac) signals of frequencies between 300 MHz
(3 x 108 H z ) and 300 GHz (3 x 10 ‘ 1 H z ). The period, T = 1If, of a microwave signal of
frequency/ , then ranges from 3 ns to 3 ps. Electrical wavelength, X = c/f, where c is the
velocity o f light in a vacuum (2.998 x 108 m/s), ranges from 1 m to 1 mm. W hat
distinguishes microwave engineering from other branches o f electrical engineering is the
above range of wavelengths. Because of the relatively short wavelengths involved,
standard circuit theory cannot be used to solve microwave network problems. In fact,
standard circuit theory is an approximation or special case o f the broader theory o f
electromagnetics, as described by Maxwell’s equations (equations (2.1) through (2.4)).
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57
Standard circuit theory relies on lumped circuit element approximations, which are not valid
at microwave frequencies. Microwave elements and components are usually described as
distributed elements, whereby the phase of a voltage or current changes significantly over
the physical length of the device, because the device dimensions are on the order of the
microwave wavelength. At much lower frequencies, the wavelength is significantly
longer, such that there is little variation in phase across the dimensions of a device or
circuit; thus the uniform field approximation is employed, and the circuit components are
treated as lumped elements. At frequencies well above the microwave frequency range, the
wavelengths are much shorter than the dimensions of the device or components, in which
case, M axwell’s equations simplify to the geometrical optics regime (Pozar, 1990).
3.2 Transmission Line Theory
At frequencies above the uniform field approximation yet below that at which antennas
can efficiently direct electromagnetic radiation, transmission lines are used. Transmission
lines are characterized by two or more conductors, typically a system of parallel plates,
parallel wires, or coaxial wires embedded in a homogeneous dielectric medium.
Schematically, the transmission line is represented by two parallel wires as shown in figure
3.1. In this configuration, one conductor is at zero potential (ground), and the other wire
supports a potential, V. In this case, conductor 1 carries a current, /, that is returned to the
source by means o f the ground conductor, 2. When V is positive, the direction o f the
electric field is from conductor
1
to conductor 2 , and the magnetic field circulates around
the wires as shown. The symbol <8 >indicates that the flux is directed into the page and the
symbol © indicates that the flux is directed out of the page. The direction of signal
propagation is in the direction of the current flow. From figure 3.1 it is clear that the
magnetic and electric fields are transverse to the direction o f propagation. The field
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58
configuration on the line defines the mode of propagation; therefore, this mode of
propagation is called transverse electromagnetic (TEM).
©
©
E
.
.
v
,
.
©
©
i
'
© © ©
© © ©
1' .. .V ’
Figure 3.1. Two conductor transmission line illustrating electric and magnetic fields
associated with TEM mode.
Although the transmission line is a distributed parameter network, the fields along the
line can be analyzed by subdividing the transmission line into an infinite number of
elemental sections, each of length Az. Each element can then be analyzed using a lumped—
element circuit model provided Az is much less than the electrical wavelength. In this
analysis, a sinusoidal source will be assumed, whereby the time variation o f the input
voltage and current is of the general form cos(cor ± <J>). However, to simplify the analysis,
the fields will be represented in their phasor or complex forms. This, as will be seen,
reduces the field functions of space and time to functions of space only. For example, a
true sinusoidal voltage source is temporarily assumed to have an e ^ tim e dependence. The
voltage source may then be represented as the product between the phasor, V = Ve±j*, and
e/tof. The phasor is introduced for convenience and the product, Ve**, is not intended to be
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59
a true source; no such eiax time dependent source exist. After the desired analysis, the true
time—domain fields are recovered by taking the real part o f the product between the phasor
and e/t“, written here as Re(v,eyovj, where Re indicates the real part of the term in the
brackets.
An approximate circuit representation of an incremental section of transmission line is
shown in figure 3.2. In the circuit of figure 3.2, R is the series resistance per unit length
for both conductors, in units of £2/m; L is the series inductance per unit length for both
conductors, in H/m; G is the shunt conductance per unit length, in S/m; and C is the shunt
capacitance per unit length, in F/m. The resistance accounts for ohmic losses along the
transmission line due to the finite conductivity of the wires, whereas the conductance
accounts for the dielectric losses in the dielectric medium surrounding the conductors.
i(z, t)
+
Viz, t)
(a)
0
/(z+Az, r)
- ... ^
—' oOo»ofl\
j
■
>
+
RAz
LAz >
-L
,+ *
v (z,t)
>GAz
CAz viz+Az, t)
------------------------S ------------ X --------0
^
^
(b)
H
Figure 3.2. Voltage and current definitions and equivalent circuit for an incremental length
o f transmission line, (a) Voltage and current definitions, (b) Lumped-element equivalent
circuit.
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60
The voltage across Az is obtained by applying KirchofFs voltage law, so that,
v(z,r) - RAzi(z,t) - LAz
ot
- v(z + Az.r) = 0
(3.1)
while KirchofFs current law leads to,
K z,t) ~ GAzv(z + Az,t) - CAz —
dt
- i(z + Az,t) = 0.
(3.2)
By dividing (3.1) and (3.2) by Az, and taking the limit as Az —>0, equations (3.1) and
(3.2) yield the following differential equations:
d v(Z’t)
r di(z,t)
— -— = -R i(z ,t) - L — - — ,
dz
dt
(3.3)
and
^
dz
= -C v (Z, , ) - C * ^ .
dt
(3.4)
So far, the previous equations have been expressed in the time-domain. The time-domain
voltage and current are now replaced with their phasor representations having an e1031time
dependence, that is, Veja* and le iw. Equations (3.3) and (3.4) now become,
d V (z)
dz
and
dz
= -(G + ja>C)V(z),
(3.6)
where the derivative with respect to time has been eliminated.By taking the derivative of
(3.5) with respect to z and inserting (3.6), and then the derivative of (3.6) with respect to z
and inserting (3.5), the following second-order differential equations are obtained:
^ p r - - f V ( z ) = 0,
dz
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(3.7)
where,
y = ^](R + j(oL)(G + y'toC) = a + /p -
(3.9)
In equation (3.9), y is called the com plex propagation constant, which is a function o f
frequency; a and f) are the attenuation and phase constants, respectively. Traveling wave
solutions to (3.8) and (3.9) are as follows:
V( z ) = V0+e-* + V0-er%
(3.10)
and
I(z) = IJe -* + I~er'.
(3.11)
By taking the derivative o f (3.10) with respect to z and inserting it into (3.5), the current is
obtained,
(3.12)
and by comparison to (3.11), the complex characteristic impedance, which relates the
voltage and current on the line, is given by,
V0+
R + ja L
V~
Zo = f r = —
= -frlQ
I
i0
(3.13;
In (3.10) and (3.11), the e r" term represents wave propagation in the +z direction, and the
eT~term represents wave propagation in the -z direction. To determine the propagation
characteristics of the true voltage and current, (3.10) and (3.11) must be converted to the
time-domain. Before proceeding, it should be noted that in equations (3.10) and (3. 1 1 ),
complex amplitudes have been assumed; that is, VJ = V^ejB, VJ = VQ~ejB,
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70+ = /0V ,
and
= / “«?*. By introducing the complex amplitudes into (3.10) and (3.11), and using
/ 0
Y = a + 7 'P, the propagation characteristics o f the true voltage and current are obtained by
conversion to the tim e-dom ain, as discussed above. It follows that,
v(z,t) = R e(V >(z)^) = Re[(V>0 +<r“V ^ + V~eazeJ^z)eJ,a‘\
= Vq e~K cos(cor —Pz + 0 +) +
c o s(gm
+
Pz +
9 "),
(3.14)
and
K z,t) = R e(/(z)e><" ) = Re[(/ 0V OK«-** +
= Ile~az cos(cot - Pz + <J>+) + /"
cos(G)r + pz + <JT).
(3.15)
Equations (3.14) and (3.15) are the real-time forms o f the voltage and current on the line.
When the loss on the line is sufficiently small, it may be approximated by a lossless
line. In this approximation, the wires are assumed to be perfect conductors (R = 0), and
the medium separating the conductors is assumed to have no associated dielectric loss (G =
0). The complex propagation constant (equation (3.9)) simplifies to,
Y = a + y'P = jc o -jL C ,
(3.16)
where it must be that a = 0 and P = co-jLC. For a lossless line there is no attenuation
because the attenuation constant is zero. It is easy to see that by putting (3.16) into (3.13),
with R = 0, ZQ =
, and is a real number. This indicates that the voltage and current
along a lossless line, o f infinite length, are in phase. The phase velocity o f propagation is
obtained by examining the velocity at a specific point (constant phase) in the wave; i.e., for
a positive z traveling wave, cot - Pz + 0+ = constant. The phase velocity of propagation is
then defined as,
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The wavelength, X, on the line, is defined as the distance between two reference points, for
example, two successive voltage maxima, or minima, at a fixed instant in time. It may also
be defined as the distance in the z direction that the wave must travel so that the phase
changes by 2 tc radians. The difference in phase can be expressed from (3.14) by,
(co r-P z + 0+)-(co r-f3 (z + A,) + 0 +) = 2 tc,
(3.18)
from which the wavelength is obtained as,
* =
(3 1 9 >
The general solutions for the voltage and current on a lossless transmission line are:
V iz) = VQ+e-* : + V~eJ^ ,
(3.20)
and
V+ n
V~
I(z ) = - ^ e - jPz—
Z0
Zq
-a
(3. 21)
The characteristic impedance of the line was given by equation (3.13), and was shown to
be governed by the incident voltage and current propagating along the transmission line.
The characteristic impedance thus depends on the line’s geometrical dimensions and
dielectric properties of the material separating the conductors. Any discontinuity along the
line or termination that differs from the characteristic impedance, therefore, requires the
alteration of the specific ratio which defines Z0. This results in reflected voltage and current
waves, a fundamental property o f distributed systems. To illustrate this property, consider
a lossless transmission line terminated in an arbitrary load impedance, ZL, as illustrated in
figure 3.3. Referring to figure 3.3, assume that an incident wave of the form
is
generated from a source located somewhere at z < 0. The ratio of the voltage to current for
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64
the traveling wave is Zq for z < 0 , however, at z = 0 , it must be that the ratio of voltage to
current is ZL. Consequently, a reflected wave must be excited from ZLof appropriate
amplitudes to satisfy this condition. At the load (z = 0) it must be that 1/(0)/1(0) = Z L.
From (3.20) and (3.21) with z = 0,
Zt =
^(0)
Vp + Vp
ho)
v *
yo —vyo~
(3.22)
Vz o
and by simply rearranging (3.22),
[T -
Z - Z± 2. T«/+
V
vo* •
Zf, + z 0
(3.23)
The complex amplitude of the reflected wave normalized to the complex amplitude o f the
incident wave is called the complex voltage reflection coefficient, and at the load is given
by.
T(0) =
Z - Z
(3.24)
L
ZL+ Z 0
In the case where ZL = Z0, r(0 ) = 0 ; that is, no reflection o f the incident wave occurs and
such a load is said to be matched.
V (0 )
z
z =0
Figure 3.3. Lossless transmission line terminated by ZL.
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65
The complex reflection coefficient can be generalized to any point on the line by
examining equation (3.10). Equation (3.10) can be rearranged to,
Viz) = K [e~ J*
] = V0 +[e-'* + f ( 0 )e'* ].
(3.25)
From (3.25) it is seen that the voltage (and current) on the line consists o f a superposition
o f incident and reflected waves; such waves are called standing waves. Equation (3.25) is
further modified to,
Viz) =
+ f (0)e2j* ] = V0+e ~ ^ [ l + f (*)],
(3.26)
where,
f (z) = f (0)e2* \
(3.27)
Equation (3.27) is the general form of the complex reflection coefficient for any point along
the transmission line. As you move along the lossless transmission line in the negative z
direction, the phase o f T(z) changes, resulting in a voltage maxima and m inim a This is
better illustrated by considering the magnitude o f (3.26):
^ ) H ^ 1 | l + n z ) |.
(3.28)
In (3.27), f(0 ) is complex and may be expressed as f(0 ) = H O )^ 70, where 0 is the
phase of the complex reflection coefficient. Equation (3.27) may now be expressed as,
f (z ) = |f(O)|e7(0+2P;).
(3.29)
Now, move from z = 0 in the negative direction to some arbitrary point -I on the
transmission line, such that z = -I. Equation (3.29) then becomes,
f ( - / ) = |f ( 0 ) |^ (e- 2po.
(3.30)
At some point -/ along the line, a voltage maximum occurs. By allowing z = -I in equation
(3.28) and inserting (3.30) into (3.28), it becomes apparent that a voltage maximum occurs
where e '(0-2p/) = l; that is, where ( 0 - 2 p / ) = 0,27C,47t,- -, and
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Similarly, a voltage minimum occurs at ej{0 2p/> = —1; that is, where
( 0 - 2 P /) = K, 3 k , 5tc, •••, and
Kni„ = \ K \ a - |r ( 0 ) |) .
The ratio o f
(3.32)
to VV,, is called the voltage standing wave ratio (VSWR) and is a measure
o f mismatch between Z 0 and Z L. The VSWR is given by,
f(0 )
1
- no)
(3.33)
The distance between any two successive voltage maxima (or minima) is X/2, while the
distance between
and Vmin is A/4.
The input impedance “looking into the line” also varies with distance X from z = 0. The
input impedance is given by,
V{-1)
Z. =
yo ?*' + f (0)e~3 ‘
em - r(0 )e -m
v +
_ l + f ( 0 )<r2^
l - f (0 )<^2y13, ° ‘
(3.34)
Using (3.24) in (3.34) yields,
z
=
in
z
tanp/
° Z 0 + yZ Ltanp/ ‘
z l + jZ Q
(3.35)
The case where ZL = ZQ(matched load) has already been examined, however, there are
other special cases worth considering. One case is where ZL = 0 or a short circuit situation.
In this case (3.24) yields T(0) = —1 and (3.35) reduces to Zin = jZ Qtanp/. Another case
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67
is where Z L = «*> or an open circuit situation. In this case (3.24) yields f (0) = 1 and
(3.35) reduces to Zin = —yZ0 cotfJ/.
The time-average power flow along the line is governed by,
Re[v(z)/(z)*]
'0
(3.36)
which shows that the average power flow is constant at any point on the line and that the
total power delivered to the load is equal to the incident power, (jv^+| /2 Z 0), minus the
reflected power, (|t^+| r(0 )| /2 Z 0). For a matched load, maximum power is delivered to
the load, i.e., there is no reflected power; for an open or short circuit situation, no power is
delivered to the load. In the more general case where the load impedance is mismatched to
the characteristic impedance, not all of the available power from the generator is delivered
to the load. This “loss” is called return loss (RL) and is defined in terms of decibel (dB)
units as.
(3.37)
A decibel is a logarithmic unit used to express power or voltage ratios. For a given power
ratio, P,/P2, conversion to decibels requires the use of the formula 10 log PJP2 (dB).
Because power is proportional to voltage squared, for a given voltage ratio, K,/V2,
conversion to decibels requires the use of the formula 20 log VJV2 (dB) (Iskander, 1992;
and Pozar, 1990).
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3.2.1 Permittivity Measurements Using Lumped Elements
From the transmission line theory just presented, it has been shown that a fundamental
property of transmission lines is wave reflection from mismatched terminations. This
property is now exploited for the determination of unknown dielectric properties in the
ffequency-domain. In the proposed technique, a small shunt capacitor, o f unknown
dielectric properties, terminating a coaxial line will be used. A requirement that the
thickness of the sample be much less than the wavelength at the measurement frequency is
also necessary in order to model the capacitor as a lumped element. The wavelength, X, in
the material medium is given by X = ^
where c is the velocity of light in a vacuum
(2.998 x 108 m/s). Figure 3.4 illustrates the coaxial sample holder and its equivalent
circuit. With no loss of generality, the complex reflection coefficient can be expressed as,
(3.38)
In this case, the load impedance is simply the capacitor reactance, Z L = - j / Co), where C
e K 'A
is the capacitance of a parallel plate capacitor, given by C = —-------= K ’C0, where A is the
area of one face of the capacitor, t is the thickness, and C0 is the geometric capacitance in
the absence of the dielectric. By inserting the capacitor reactance into (3.38),
(3.39)
By expanding (3.39) and consolidating real and imaginary parts, it can be shown that,
(3.40)
and
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69
K"
(3.41)
C»CoZ0 (|f(O ) | 2 + 2 |f (0)|cos9 +1)
The method therefore requires the measurement of the modulus and phase angle of the
reflection coefficient o f the sample holder and sample configuration, at the desired
frequency, provided that the geometrical dimensions o f the capacitor are much smaller than
the wavelength, and the use o f equations (3.40) and (3.41) to calculate the complex
permittivity. By rearranging (3.39) it can also be shown that,
f (0)| = \ l ~ 2 K " ( Z 0/ Z c) + (Z0/Z c)2( K '2 + K " 2) T
I [ i + 2 * " ( z 0 / z c) + ( z 0 / z c ) 2 ( * : ' 2 + * :''2)J ’
(3.42)
and
0
= arc tan
(3.43)
where Zc = l/o)C0 is the impedance of an empty capacitor. For a K ' > 1 and a K " > 0, the
modulus of the reflection coefficient changes between 1 and 0.4142, and the phase angle
changes between 0 and 180° (Iskander and Stuchly, 1972; and Stuchly, et al., 1974).
Sample
Coaxial
Connector
I
H
ZcP
-/
— Zc=j(ofCQ
I
z=
0
Figure 3.4. Illustration of coaxial sample holder and equivalent circuit.
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70
3.2.2 Error Considerations
W hen the estimated uncertainties in the measurement of the phase angle and modulus of
the reflection coefficient are approximately equal, that is, A0 = A
the
uncertainties in K ' and K " reduce to,
AK '
K'
ACn
\ 2
1/2
AZ0
+ (A 6
(3.44)
) 2
and
1/2
AK "
K"
ACn
C
V '-'0
AZ„
+ (A0 ) 2
V Zo
I
tan
(3.45)
8
The condition for the minimum uncertainty in K ' is achieved when the following ratio is
maintained:
= [K '1 + K " z]ir2 = K .
(3.46)
Equation (3.46) allows for the selection of the optimum capacitance of an empty sample
holder at a specific frequency. This capacitance, C0, then fixes the optimum thickness o f
the sample, at a specific frequency. Equation (3.45) shows that the uncertainty in K "
decreases for lossy materials. Generally, this method is more suitable for lossy dielectrics
(tanS > 0 . 1 ) and is less strongly dependent upon selection of the “optimum” thickness
(Iskander and Stuchly, 1972; and Stuchly, et al., 1974).
Finally, the effect of fringing field on the permittivity measurements is considered. In
this case, the total capacitance is taken as the sample capacitance in parallel with the
capacitance due to the fringing field, Clol = K 'C0 + Cf . By using Ctot in (3.38) and
solving, it can be shown that,
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From (3.47), it is clear that the effect of the fringing field is to increase the measured AT',
while AT" is unaffected and remains governed by (3.41). The fringing field contribution is
expected to increase with low K dielectrics because in higher ATdielectrics, the electric field
lines tend to concentrate more in the dielectric, thus reducing the fringing effect (Iskander
and Stuchly, 1978).
3.3 Wave Propagation Along Circular Cylindrical Waveguides
Another way of transmitting microwave power is by means o f a waveguide.
Waveguides generally consist of a single, closed conductor o f some uniform cross-section.
In this section, wave propagation along an air-filled circular cylindrical waveguide will be
considered. The general approach to the analysis of other waveguides of different crosssectional geometries is exactly the same. Next, a resonant cavity is constructed by applying
the short circuit boundary conditions at the waveguide ends. The change in resonant
properties of the cavity when it is partially filled with a dielectric is examined, which leads
to a method for the measurement of material properties at microwave properties. Then, the
resonant behavior of dielectric waveguides are examined, and is related to the dielectric
properties of the material. Finally, the effects of filling the cross-section of a small length
of a waveguide, with some material, are discussed, and again used for the determination of
dielectric properties.
First, M axwell’s equations ((2.1) to (2.4)) are converted to their phasor or complex
vector form. This is accomplished by replacing the sources and fields in (2.1) through
(2.4) with sources and fields which possess complex time variations; that is, J = Jem ,
p = peja*, E = EeJW, and B = BejW. Maxwell’s equations then become,
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72
V e0E = p,
(3.48)
V B = 0,
(3.49)
V x £ = -jo&B = -j(O[L0H ,
(3.50)
and,
V x H = J + j(aeQE ,
(3.51)
B
where H = — , and is defined as the magnetic field intensity, in units of A/m. In a source
M-o
free environment, J - p = 0, and (3.48) and (3.51) are modified as follows:
V e 0E = 0 ,
(3.52)
and
V x H = jG)EQE .
(3.53)
The circular waveguide is essentially a cylindrical hollow metal tube of circular crosssection capable of supporting wave propagation along its length. The system is thus
characterized by a single conductor, as opposed to the transmission line, which was
defined by two parallel conductors of some configuration. Single conductor waveguides
cannot support TEM waves, however, they can support what are called transverse electric
(TE) and transverse magnetic (TM) waves. The geometry of the circular waveguide
defined in terms of a cylindrical coordinate system is shown in figure 3.5.
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Figure 3.5. Geometry of circular cylindrical waveguide.
By assuming time-harmonic fields with an ^ “ dependence, the phasors H and E can be
represented as,
E = Epap + E ^ + E.a. = (Epap + E ^ + E,a, )e~®z,
(3.54)
and
H = Hpap +
+ H.a, = (H pap +
+ H.a. )e“7p\
(3.55)
for an electromagnetic wave propagating in the positive z -direction, in a lossless medium.
For a general vector A = Apap + \ a ^ + A,a, in cylindrical coordinates,
^
1 3A.
V x A = (—
p 30
dA.
5A> ^A.
-^-)a f + ( ^ dz
dz
dp
1 5(pA4) 3Aa
- -£*•)&.
p
3p
30
(3.56)
Using (3.56), (3.53) yields three equations:
(357)
1*
and,
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(3.58)
Also, by using (3.56), (3.50) yields three equations:
1 dE
dE*
1 dE
paf“i r =piT+Jp *='yo*°Hp'
dE
dE,
dE,
l k ~ ~ d p ~ ~ J^ Ep~ ~ d ^ ~
<3-60)
(3'61)
and,
1
d (p E .)
dE
p
dtp
dtp
) = -M io ^ •
(3.62)
Using the system of equations (3.57) through (3.62), the transverse components o f the
electromagnetic wave traveling down the waveguide can be derived in terms o f the
longitudinal components as,
«<. = T T (P dp
x i + —p
K =
p 90
90
“ COPo ^ 0 ,
dp
ir =
- 7J2 (—
(^ o ^
Ho
k 2 p 90
dp
( 3 M)
(3.64)
(3-65)
and,
( 3 -6 6 )
In equations (3.63) through (3.66), the wavenumber k = O3 ^/p 0 e0 , such that
(co2 p 0 e 0 - p 2) = (k 2 —p 2) = k] , where kc is the cutoff wavenumber. The meaning o f kc
will soon be made clear. By taking the curl of (3.50) and (3.53) it easy to show that,
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75
V x (V x E) —k 2E = 0 ,
(3.67)
and
V x ( V x H) —k 2H = 0.
(3.68)
Equations (3.67) and (3.68) are called the vector Helmhotz equations and to find the fields
present in the waveguide the strategy isto look for solutions to (3.67) and (3.68).
For TE modes the components of theelectric field are transverse to the z -direction of
propagation, therefore, E. is equal to zero, with H . * 0 . Equation (3.68) can then be
expressed as,
(■ fj +~
+ 4 - ^ - ^+ k l )H. = 0.
8 p
p 3p p 9 < )) 2
(3.69)
The method of separation of variables may be used to solve (3.69) by letting
H X p,z) = /?(p)P(<j>), so that (3.69) can be rearranged to,
p 2 d 2R p dR
2l2
-1 d 2P
~— 7 T + — + P kc =
rR dp
R dp
P d§
(3.70)
Since the left side of (3.70) depends only on p and the right side o f (3.70) depends only on
<}>, each side may be equated to a constant, so that,
-1 d 2P
= constant = k 2
P d§ 2
(3.71)
or,
d 2P
4-fc2P = 0 .
d *>2
♦
(3.72)
By using (3.71) in (3.70), equation (3.70) may be rewritten as,
, d 2R
dR
, ,
,
P -T T + P — + (P
- ^ ) / ? = 0.
dp
dp
The general solution to (3.72) is,
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(3.73)
P(<J>) = A sin fyj) + BcosA:^.
76
(3.74)
Since the solution to H. must be periodic in <f>, k^ must be an integer n, thus (3.73)
becomes,
h'
d 2R
dR
. 2 i-p — + ( p % - n 2)R = 0 ,
dp
dp
(3.75)
which is recognized as Bessel’s differential equation. The solution is,
R( p) = CJn(kcp) + DYn(kcp),
(3.76)
where Jn(kcp) and Yn(kcp) are the Bessel functions o f the first and second kinds,
respectively. In (3.76) D must be set equal to zero otherwise for p = 0, Yn(kcp) would
become infinite. The solution to H. may now be written as,
H. = /?(p)P(<{>) = (A sin «<|>+ B cos/«{>)/„ (A:cp),
(3.77)
where the constant C has been absorbed in the constants A and B. Remembering that for
TE modes Ez = 0, and using (3.77) (that is, H, = H . e '^ ) in equations (3.63) through
(3.66), the transverse components o f the fields are obtained as follows:
Ep =
(Acoswj) - B sin nty)Jn(kcp)g~yfe,
(3.78)
KP
E* = ^ p 2 -(Asinn<|) + Bcosnty)J'a(kcp)e~jfk,
(3.79)
kc
= -/^(Asinn<|> +£cos/z<(>)./'(A:cp)e_yP\
(3.80)
K
and,
= - 7 j^(Acos«<(>-fisinn<|>)/n(fccp)e-^ \
kcP
The wave impedance is defined as,
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(3.81)
Finally, to determine kc, the boundary condition that the tangential component of the electric
field, Ejj,, = 0 at the waveguide wall (p = a) must be enforced. Since already by definition
a
of a TE mode, E, = 0, it must be that the other tangential field component,
= 0 for p =
a. Applying this condition to (3.79) yields J '(k ca) = 0 . If the roots o f J '( x ) are defined
as pr nm
' 7 so that Jti '' i( pn m' ') = 0,7 where rp 'run is the mth root o f Jn '7 then k c must have the value,*
/
k
*"
=
(3.83)
a
Values of p'nm are given in mathematical tables; the first few values are given in Table 3.1.
The TE„m
nm modes are thus defined by
J the cutoff wavenumber, kr
c llm ’, where n refers to the
number of circumferential (<j)) variations, and m refers to the num ber of radial (p)
variations. The propagation constant o f the TE^, mode is given by,
P- =
=J*
V
1
-
f p’
—
•
v a /
(3 -84>
with a cutoff frequency of
= ~ kr
fc
"
= „
-?
2nj\L0£0
2najiiQ
£Q •
(3.S5)
Below the cutoff frequency the modes are highly attenuated (evanescent modes) and
effectively do not propagate.
Table 3.1 Values of p '^ for T E ^ Modes of a Circular Waveguide
n
0
1
2
P
hi
3.832
1.841
3.054
Pn2
Pn3
7.016
5.331
6.706
10.174
8.536
9.970
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78
For the TM modes the components o f the magnetic field are transverse to the z-direction
o f propagation, therefore, H, is equal to zero, and Ez * 0. Equation (3.67) can then be
expressed as,
(|dpt + -p |dp
- + -p T
^j>- + ^ ) £ '= = 0 ‘ d<
(3.86)
The method of solution is completely analogous to that for the TE modes, leading to the
general solution,
E. = (Asin/i<J> + Z?cos«<t>)/n(&cp).
(3.87)
In this case, however, the boundary conditions may be applied directly to E, given by
(3.87), that is, since E.(p, <)>) = 0 at p = a, it must be that J„(kca) = 0, thus,
= — .
a
(3.88)
Values of pnmare given in mathematical tables; the firstfew values are given in Table 3.2.
The propagation constant of the T M ^ mode is given by,
P™, = V *
2 - *
c2
=
.
(3.89)
with a cutoff frequency of
f
= ----c"
Ic
D
= ------tlnm----2n a ^ iiQ£Q
n
9 0
Remembering that for TM modes H. = 0, and using (3.87) (that is, E. = Eze~3z) in
equations (3.63) through (3.66), the transverse components of the fields are obtained as
follows:
Ep = -^ (A sin « < |) + flcosn<j>)./'(Acp)e-y'pz,
(3.91)
kC
E^ = ^ P ” (Acosm|> - fisin /!$)/,,(fccp)e-y1J\
k CP
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(3.92)
)
79
■/(0 £on
Dc:n nlK\ T f i r
Hp = —
j-2—/(Acosn<j>-flsinnty)Jn(kcp)e
' ,
kcP
(3.93)
and,
H = ~-/C0£q (Asinmf) + flc o s w tO /^ p )* " '* .
(3.94)
fcc
The wave impedance is,
9
<395)
P
Table 3.2 Values of p nm for T M ^ Modes of a Circular Waveguide
n
0
1
2
P n l
2.405
3.832
5.135
5.520
7.016
8.417
P n l
8.654
10.174
11.620
In the above solutions, ((3.77) to (3.81), (3.87), and (3.91) to (3.94)), there are two
remaining arbitrary amplitude constants, A and B. These constants control the amplitude of
the sin/z<j> and cos«<j) terms, each o f which are independent. Because of the azimuthal
symmetry of the circular waveguide, both the sin/i<J> and cosn<t> terms are valid solutions;
however, the actual amplitudes of these terms will be dependent on the excitation of the
waveguide. From a different point o f view, the coordinate system can be rotated about the
z-axis such that either one of the amplitude constants o f the longitudinal field is zero. By
this argument, one of the amplitude constants, A or B, may be set equal to zero, therefore,
in subsequent analysis, A will be set equal to zero (Pozar, 1990; Argence and Kahan,
1967; and Kajfez and Guillon, 1986).
3.3.1 Circular Waveguide Cavity
A resonant cavity is constructed from a section of circular waveguide by shorting both
ends. The fields within the hollow, cylindrical waveguide resonator must obey Maxwell’s
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80
equations for source free regions. It will be shown that the waveguide resonator has many
resonance frequencies associated with the various field distributions or modes within the
cavity and that electromagnetic fields cannot be sustained within a lossless cavity except at a
resonant frequency. The geometry of a cylindrical cavity is shown in figure 3.6.
Figure 3.6. Geometry of cylindrical cavity.
A solution for the resonant modes and associated frequencies could be obtained by
applying the appropriate boundary conditions to Maxwell’s equations, however, the
solution is simplified by beginning with the circular waveguide modes already satisfying
the boundary conditions at the waveguide walls. By shorting the ends o f the waveguide it
is obvious that standing waves must form in the cavity. With this in mind, the transverse
electric fields for either the TE or TM mode may be expressed by a single equation,
E,(p,<M) = e(p,$)[A+e~®’1"z + A~efimZ],
(3.96)
where e(p,<|>) represents the transverse variation o f the mode, and A+and A are arbitrary
amplitudes of the forward and backward traveling waves. The propagation constants for
both TE m and TMnmmodes were given in equations (3.84) and (3.89). Next, the
additional boundary requirements are introduced, that is, at z = 0 or d, Et — 0. For z = 0,
i
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81
(3.96) yields A+= -A'. W hen this substitution is made, and the boundary condition that
Et = 0 at z = d is imposed, (3.96) reduces to,
^ +sinpnmJ = 0,
$md = ln , for / = 0 ,1,2,3,4,—.
(3.97)
By rearranging (3.97), P ^ = In I d . The guide wavelength is given by,
In
run
=
2d
(3-98)
*
From (3.98), d = /(—A.^), which says the cavity must be an integer number o f half
wavelengths long. The resonant frequency for either mode is obtained from (3.84) and
(3.89). To obtain the resonant frequency for the T E ^ mode, rearrange (3.84) so that,
k 2 = <o2 p 0 e 0 = PL + (— )2a
(3.99)
It follows that,
= 2nf^]yL0£ 0 = ^ P L + P ^ ) 2 ,
(3.100)
from which,
?TE _
fnZi
=
K \ 2 ,(Pnm
~ Jr - — J (/ —
)~ + ( ^ )
2 n ^ ii 0e 0 \ d
a
2
_=
C M v2 , , P'nmn2
—
( - f + ( i^ L ) 2 ,
2n V d
a
(3.101)
where c is the velocity o f light. W hen the cavity is filled with some material having relative
permeability p r and relative permittivity er, (3.101) becomes,
fnmi —----- 7=
J( ~ )
2 n^jiLr£r V d
2
+ p ^ - ) 2 . In an identical manner, the resonant frequency for the
a
T M ^ mode is obtained, and is as follows (Pozar, 1990):
f™ = „ r
J P ) 2 + ( — )2 = p J P )2 + ( — ) 2 ,
"""
2 n^]\i 0£ 0 v d
a
2n \ d
a
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82
1
Jtmm
/ w = 2. 7 1-c—
^ 1 ^ ,V/;( "d j ) 2 + (EssL)2
a
(fllIedcavity}
(3 -102>
3.3.2 Quality Factor ( 0 and Field Configuration of TM , ) I 0 M ode
Near resonance, a microwave resonator may be modeled in terms of either a series or
parallel RLC lumped-element circuit. By examining the complex power delivered to the
resonant circuit, the characteristics of resonators are developed. Consider the parallel
resonant circuit shown in figure 3.7.
I'V*) v
Figure 3.7. Parallel RLC resonant circuit.
The input impedance is given by,
z" = G
(3.103)
r i +H
■
The complex power delivered to the resonator is.
2
p .., =
- v i '
2
=
- z
2
m \i\~
11
= - \ v
2'
1
Z
.-.
. = H V 2f
1
1
2
1
R
+. J
coL
ja C I.
(3.104)
The power dissipated by the resistor, R, is
P
* /« v
= —
2 R
(3.105)
the average electric energy stored in the capacitor, C, is
w-
- jM!c-
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(3.106)
83
and the average magnetic energy stored in the inductor, L, is
<3107>
=
where I L is the current through the inductor. Using (3.105), (3.106) and (3.107), (3.104)
can be rewritten as,
K ^ P , m +i<s>2W„-ja,2W, = Pu>„ + 2j<M.Wm- W ,) .
(3.108)
Using (3.108) the input impedance can now be written as.
,
_ 2P„ _ P<„,+VoXWm - W ,)
in ~
|„i2 —
«,2 /
/
/
/
(3.109)
2
2
At resonance, the averaged stored magnetic energy equals the average stored electric
energy, Wm = We. From (3.109), the input impedance at resonance is,
Z ..=
p
'2oss = R ,
(3.110)
12
and is purely real. By equating (3.106) to (3.107) and solving for to, the resonant
frequency is found as,
" • ’ T b c-
<311I)
An important parameter in the description of resonant circuits is the Q, or quality factor,
which is defined as,
Q _ m (average energy stored) _ ^ Wm - We
(energy loss/second)
Ploss
(3 . 1
1 2
)
The Q is thus a measure of the loss of a resonant circuit. As loss decreases, Q increases.
At resonance, by use o f (3.111) and (3.107), (3.112) becomes,
d
e = G)0— ^ = — r = <o0 /?C.
P lo ss
® 0L
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(3.113)
84
To understand the behavior of Z-m near the resonant frequency, co in (3.104) is replaced
with co0 + Aco. Equation (3.104) then becomes,
v-l
Z. =
— + --------------------------+ /‘(G )0
R
j((o 0 +Ao})L
I
R
+Aco)C
0
=
)
• AtO
\
y
+ y — — + yAtoC + jco0C
0 )qL
- + - —AQ^ (0° + yco0C + y'AcoCj
R
joa0L
v ‘
, -i
— 4 - y - ^ - + yAcoC
R J <&\L
j
-i
— i- /Aco
R
+C
J)
= | — + 2 yAcoC1
' R
J
= ------- - ------1 + 2 jAo&RC
(3.114)
By using (3 .113), i.e., RC = £?/co0, in (3.114), the input impedance becomes,
Z.„ =
(3.115)
I + 2y'£>Aco/co0
Figure 3.8 shows the behavior of the magnitude of the input impedance versus frequency.
0 .7 0 7 /t
BW
Figure 3.8. Input impedance magnitude versus frequency.
The half-power bandwidth edges occur at frequencies (Aco/co0 = BW/2, where BW is the
fractional bandwidth) such that |Z i n | 2 = R 2/ l , which implies that, from (3.115)
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85
(3.116)
The Q of the resonant circuit itself, in the absence o f any loading effects caused by some
external circuitry, is called the unloaded Q, Qu. In most situations, the resonant circuit is
coupled to some external circuitry. This external circuitry may have associated with it
some impedance which combines with the resistance of the resonant circuit, contributing an
external Q, or Qe, the effect o f which, m ay lower the overall Q, or loaded Q, designated QL.
The Qs of the resonant circuit and any external effects combine as follows:
— = — +— .
Q l
a
(3.117)
a
The fields of the T M ^ mode are obtained as follows: from equations (3.91) and (3.92)
(with A = 0) in (3.96), and remembering that the boundary conditions at the waveguide
wall resulted in A += -A', Ep and Ep are obtained. Recalling that H . = 0 , equations (3.57)
and (3.58) yield
and Hp. Finally, (3.65) yields Ez. The fields o f the T M ^ modes are
thus,
H. = 0 ,
(3.118)
(3.119)
p)sin /i<t>sin— z,
Jn
Pnm
a
a
p) sin n<(>c o s— z ,
d
p)cos/z<|>cos— z,
Ez = £ 0 7n(-^!!2 -p)cosn<t)cos— z.
a
d
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(3.120)
(3.121)
(3.122)
(3.123)
86
In equations (3.118) to (3.123) E 0 = 2A*B, and coe0 = k^Je.Q/ |i 0
have been used. The
TM„io (n = 0 , m = 1 , 1 = 0 ) mode is the fundamental mode, that is, the resonant mode o f
lowest frequency provided the condition dJa < 2 is maintained. In this case the above fields
reduce to zero except for E. and
. The field configuration o f the T M ^ mode is shown
in figure 3.9 (Pozar, 1990; Argence and Kahan, 1967; and Kajfez and Guillon, 1986).
t
I
3.9. Electric and magnetic fields of TM q^ mode. Flux density is proportional to field
intensity.
3.3.3 The Cavity Perturbational Method
The resonant properties o f the cavity resonator may be modified by introducing a small
material perturbation into the closed cavity. Thus, this provides a method for the
determination of the dielectric properties o f materials by measuring the shift in resonant
frequency. The method assumes that the actual fields in the perturbed cavity are not
dramatically different from those in the unperturbed cavity. To begin with, define E 0 and
•<
_
_
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ho as the fields in the original unperturbed (empty) cavity. Define E and H as the fields
in the perturbed cavity. Now, write Maxwell’s curl equations for both the unperturbed,
Vx
£ 0
= -y'coof±/70
(3.124)
V x H 0 = j(O0eE 0
(3.125)
and perturbed,
V x E = - j(o(iL + A\l )H
(3.126)
V x H = ;o)(e + t e ) E
(3.127)
cavity, where co0 is the resonant frequency of the unperturbed cavity and co is the resonant
frequency of the perturbed cavity. Using equations (3.124) through (3.127) it can be
shown that,
- l ( * z E E ' o + * \ L H H ' 0)dv
— -------= y ~1 :
- -.x
,
coo-rn
(3-128)
\ Va( e E - E ; + i L H H ; ) d v
©
where V0 is the volume of the cavity. Equation (3.128) is an exact equation for the change
in resonant frequency due to material perturbations. The problem with (3.128) is that the
exact form of E and H is not known. However, by assuming that Ae and Ap are small,
A.
A
A
the fields E and H , may be approximated by the fields, E0 and H 0 . Assuming the
material is non-magnetic, is of the shape of a long thin rod of height equal to the cavity, of
uniform cross-section, and is placed in the center o f the cavity such that its principal axis is
parallel to E,, (3.128) becomes,
2
C0 n
<*>0
=^
dv
(3.129)
fo
}vc |4 |
2 jv
In (3.129), Vs and Vc indicate the integrals are taken over the sample volume and cavity
volume, respectively. This sample configuration is particularly convenient, because, as
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88
shown in figure 3.9, a thin sample is positioned in an approximately uniform electric field.
Using the fields for the T M ^ mode (3.129) becomes,
V Af
K ' = 0.539— — + 1 ,
V. fo
(3.130)
and
K" =
0.539 V J
2 K ' Vs Q
1
_
(3.131)
Qo
There are no restrictions on the magnitude of the dielectric constant and dissipation factor,
insofar as Aflf 0 remains very small. By (3.130), Af/fQis maintained small if Vs is kept
small; that is, the cross-sectional area of the sample must be kept as small as possible
(Pozar, 1990; Argence and Kahan, 1967; and Nakamura and Furuichi, 1960).
A variation of the cavity perturbation method whereby the sample is less than the height
of the cavity is introduced here as well. Following Parkash et al., the sample is treated as a
dipole equal in length to the sample. Using the method o f images, the effects of the
polarization of the sample and its image dipoles on the net depolarizing field in the sample
are considered, yielding an effective depolarizing factor,
..
nh
nh
N„ = N
cot2H
2H
(3.132)
where N is the depolarizing factor and depends on the axial ratio of the specimen (Bozorth
and Chapin, 1942), 2H is the height of the cavity, and 2h is the length of the sample. By
considering the fields in both the unperturbed and perturbed cavity, the dielectric
parameters are obtained as,
Sco
v0
K '~ 1 =
0 )n
5(0
~ N eV0 —
_2Jt{ka)
0 ),o .
2
- N
e v 02
1
2
- N ,K —
(0 n
8[ —
2
0
-
11
UeJ
+ N eV0 8 f
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(3.133)
i
89
f1 -2 Q1 J r_ 2 y,2^(to )
K" =
+ N X — s( - )
“o U eJ
8
A FvH
Ci>„
(3.134)
2
2
- v>
U '-(k a )
where
NV
+
] represents the difference
2Q )
2
{2Q )_
'j ~
2
i
Qo are the quality
‘
factors of the filled and unfilled cavity, respectively, tD0 is the resonant frequency of the
unperturbed cavity,
8
® is the difference between resonant frequencies for the unperturbed
and perturbed cavity, and ^ 7 , 2 (yfeat) is 1.8552 for first-order mode excitation. In (3.133)
and (3.134), V, and V0 are the sample and cavitiy volumes, respectively. When the sample
is allowed to equal the height of the cavity, (3.133) and (3.134) reduce to (3.130) and
(3.131) (Parkash, et al„ 1979).
3.3.4 Shorted Dielectric Post Resonator
A right circular cylindrical dielectric rod, o f low loss and high dielectric constant also
behaves as a resonator, similar in principle to the cavity resonator just discussed. The
high-dieleclectric constant of the resonator ensures that most o f the fields are contained
within the dielectric; however, unlike the metallic cavities, there is some fringing or leakage
from the sides and ends o f the resonator. Another method for the measurement o f the
dielectric properties of materials is now discussed. In this method the dielectric resonator is
shorted at both ends, and its resonant spectra analyzed. Again beginning with M axwell’s
equations and imposing the appropriate boundary conditions, it can be shown that for an
isotropic cylindrical sample o f radius a and length L placed between two infinite parallel
conducting plates resonating in the TE0n/ mode, the dielectric constant may be obtained
from the following transcendental equation:
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90
rc /p (a ) _
7, ( a )
Q ^O(P)
/sf.CP)’
(3.135)
where 70 ( a ) and 7, (a ) are the Bessel functions of the first kind of orders zero and one,
respectively; AT0 (p) and AT, ((3) are the modified Bessel functions of the second kind of
orders zero and one, respectively. The parameters a and |3 are as follows;
2
ita
a = ----- K K0
*
2
P=
2 iza
f
c
(3.136)
\
V
p J
.
\1
-i
(3.137)
s y Pj
where c is the velocity of light, and vp is the phase velocity in the dielectric medium and
c
( IXo_
2L
(3.138)
In (3.138) / is the number of longitudinal variations o f the field along the z-axis and A. 0 is
the free space resonant wavelength. To each value o f (3 there is an infinite number of
solutions a „ . Consequently, for a dielectric rod resonating in the TE0n/ mode, to each (3,
given by (3.139),
2 na
a.
2L
1 /2
- 1
(3.139)
there corresponds an a n. Knowing the indices n and /, the dielectric constant is given by,
\2 . ( I K
K =f
v0 ' 2
I 2 na
2L
(3.140)
The dissipation factor is obtained by analyzing the unloaded Q of the dielectric post, and is
given by,
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91
B,
tan 8 =
(3.141)
a
where
A = 1+
^ o (P /)^ 2 ( P / ) - ^ . 2 (P,)
KK2{p;) 7,2 ( a n) - 7 0 ( a n)y 2 ( a n)_
(3.142)
and
B=
K ^ K ^ - K 2^ )
l 2K
1+
2 K f Y K z 0l} | / ‘ K f{p,) / I2 ( a n) - y 0 ( a n)7 2 ( a n)
(3.143)
where Rs is the surface resistance o f the end walls.
For high K materials, (3.141) may be approximated by
tan 8 = — ,
Qu
(3.144)
where Qu is the unloaded quality factor. For typical sample dimensions, the TE0I, mode is
dominant, and easiest to couple into; consequently, this is the mode of choice for
measurement of relative permittivities. The field configuration o f the TE0I, mode for a
shorted dielectric post is shown in figure 3.10.
Metal Plate
ff—field
Sample
Figure 3.10. TE0II field configuration in shorted dielectric rod resonator.
The method is highly accurate in the determination of AT(± 0.1 per cent) but somewhat
restricted in the determination of tanS < 5 x
1 0
“\ because of the uncertainty in the actual
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value o f Rs. The method, however, is dependent upon the observation and correct
identification o f the appropriate resonant mode. Kobayashi and Tanaka conducted a
generalized study o f the resonant modes and developed a generalized mode chart (figure
3.11) for the enumeration o f modes, including cutoff conditions. Using the chart, given
known DIL ratio, where D is the diameter of the post, the order in which the resonant
modes appear are obtained by simply moving up the chart at the appropriate (DIL)2 value
(Kajfez and Guillon, 1986; Kobayashi and Katoh, 1985; Kobayashi and Tanaka, 1980;
Courtney, 1970; and Hakki and Coleman, 1960).
cutotl eo n d io n tor !•!
TM
Figure 3.11. Mode chart for a dielectric rod resonator short-circuited at both ends
(Kobayashi and Tanaka, 1980).
3.3.5 Filled Coaxial lines and Waveguides
In this method, a material is shaped to completely fill the cross-section of a length of
transmission line or waveguide. The change in impedance and propagation characteristics
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93
in the loaded section of the Line or waveguide is used to calculate the dielectric properties o f
the material. The method may employ either coaxial transmission lines or waveguides of
some uniform cross-section as shown if figure 3.12.
Dielectric
Rectangular W aveguide
Coaxial W aveguide
Figure 3.12. Partially filled rectangular and coaxial waveguides.
The method of analysis is essentially identical regardless of the line configuration, and
basically involves the measurement of the scattering parameters of the filled line or
waveguide. Scattering parameters are a convenient way of analyzing microwave m ulti-port
networks, with the scattering matrix providing a complete description of any Af-port
network. The measurement method currendy being discussed is described as a two-port
network. A flow diagram for a two—port network is shown in figure 3.13. The [S] matrix
is defined in reladon to incident and reflected voltage waves as follows:
=5
sm
SiN y +'
:
V2+
I
•
Sm_
1
ya.
•••
-----
S2l
I
512
1
Su
V
V2~
*
•
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.145)
94
where a specific element of the [S] matrix can be determined as
V
S =—
'> vi +
(3.146)
’
Equation (3.146) says that S(>is found by measuring the reflected voltage wave amplitude
at port i and dividing it by the incident voltage wave amplitude measured at port j , with all
other ports terminated with matched loads.
$ 2 1
>
S 22
5 l l >f
--------^ ------------- ------- <
---------- ------- < ----------
^ 1 2
Figure 3.13. Signal flow graph representation of a two-port network.
Again by solving the system o f Maxwell’s Equations with appropriate boundary
conditions, it is possible to relate the scattering coefficients S,,(o)) and S2 l(to) to the
reflection coefficient ( T) and transmission coefficient (7) with the following equations:
S. .(CD)
=
( i - r 2) f
i - r 2r 2 ’
(3.147)
d - r 2)7
i - r 2f 2
(3.148)
S,.(CD) =
r =
(£
IK
K'
- 1
(3.149)
+ 1
and
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95
T = e x p j—y(co/c)^J[i'rK ’ d}.
(3.150)
In the above equations T is the transmission coefficient and d is the sample thickness.
Equations (3.147) and (3.148) allow for the determination o f f and 7, by measuring the
scattering parameters, from which the complex permittivity is calculated. For a filled
rectangular waveguide, and from (3.147), (3.148), (3.149), and (3.150), it follows that,
r = X ± a/X 2 —1, where X
(3.151)
ii
T
f a , + s 2, } - f
1 - { S „ + S 2l} f ’
(3.152)
and
X, where
(3.153)
Several errors associated with the use of this method contribute to its overall lowered
accuracy as compared to the other techniques discussed. These errors include dimensional
errors caused by air gaps between the material and conductors, by surface roughness, by
thickness variations of the filling material, and possible excitation of higher order modes.
The modeling of all these error contributions is rather complex, however, errors > ± 1 per
cent are expected in the measurement o f dielectric constant, with the technique being
unsuitable in the accurate measurement o f tanS < 0 . 1 (Nicolson and Ross, 1970; and
Hewlett Packard, 1985).
3.4 Summary of Measurement Techniques
Several broadband and resonant methods for the determination of the dielectric
properties of materials have been discussed. A summary of these techniques as well as
their limitations is presented in Table 3.3.
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96
Table 3.3 Summary o f Microwave Dielectric Measurement Techniques
Technique
Lumped Element
Cavity Perturbation
(TMoIOcavity)
Post Resonator
Filled waveguide
Comments
sample geometry:
small electroded disks
High—ATresults in
impedance mismatch;
hard to calibrate
higher order
propagating modes
possible
uniform field
approximation breaks
down at higher
frequencies
possible fringing field
effects
Accuracy
Frequency
Broadband
(50 MHz to < 1
GHz) depends on
K
sample geometry:
long thin rods
AflfQmust be kept
small
Fixed frequency
sample geometry:
right cylindrical rod
must be high Q
correct mode
identification required
Fixed frequency
sample geometry:
must completely fill
cross-section of
waveguide
Sample configuration
leads to multiple
sources of errors
higher order mode
propagation possible
Broadband
(> 500 GHz)
(> 1 GHz)
(dependent upon
waveguide
geometry)
limited, < ±
cent
1
per
100
tanS > 0 . 1
high, generally
within ± 1 per
cent. Error due to
high Aflf 0 (> 1 %)
and sample
dimensions
very high (± 0 . 1
per cent) for K,
limited for tanS <
5 x 10-4. Error
due to Rs and
sample
dimensions
limited low loss
resolution
tan5 > 0 . 1
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CHAPTER IV
REVIEW OF LITERATURE
This chapter presents a historical overview of observed high-frequency dielectric
relaxation in polycrystalline ceramic, polydomain single crystal and single domain single
crystal barium titanate (BaTiOj). In the initial sections, theories which either explained or
predicted high-frequency relaxation in ferroelectrics are presented. In the subsequent
sections, observations made on pure B aTi0 3 will be emphasized, however, relaxation
phenomena observed in other alkaline earth titanates, their solid solutions with B aT i03, and
other ferroelectric compositions will be briefly discussed whenever appropriate.
4.1 Theories for Dielectric Relaxation in Ferroelectrics
Earliest attempts to explain the high-frequency relaxation in ferroelectric BaTi0 3 were
made by Mason and Matthias (Mason and Matthias, 1948). In their model of B aTi0 3 the Ti
ion alternates between a number of equilibrium positions, suggesting the existence o f some
natural relaxation frequency associated with its movement. By making specific
assumptions about the size of the potential well separating Ti equilibrium positions, the
relaxation frequency was placed in the microwave frequency region. The problem with this
model, however, is that it relies on an ionic polarization mechanism, which as discussed in
chapter n, would likely exhibit resonance at optical frequencies.
Devonshire (Devonshire, 1951) suggested that the high-frequency relaxation
phenomenon was due to piezoelectrically active domains. In an unpolarized ceramic
individual grains consist of polarized domains oriented in various directions, consequendy,
the sample is macroscopically piezoelectrically inactive. In an alternating field there is no
vibration of the sample as a whole, however, the domains remain piezoelectrically active,
whereby individual or coupled groups of domains can vibrate. Thus, it is expected that the
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98
effective resonance frequency will be approximately equal to that of a single domain. Since
resonance frequencies are inversely proportional to linear dimensions, domain widths on
the order o f microns should yield resonance frequencies in the microwave frequency
region.
Kittel attributed the high-frequency relaxation in B aT i0 3 to the inertia of 180° domain
walls (Kittel, 1951). The surface energy o f the boundary will be increased when it is set in
motion because of the inertia o f the ions which change position slightly as their dipole
moments reorient on passage of the domain wall. The surface inertia of the 180°
boundaries in B aTi0 3 was derived from a simple model, having a boundary N lattice
constants thick (separating oppositely polarized domains). The principal mechanism giving
rise to this inertial effect is the displacement o f Ti ions, which are assumed to change
position within the unit cell by a distance o f 5, on passage of the wall. In uniform wall
motion the average velocity of the Ti ions, vn , is related to the wall velocity, vw, whereby
V Ti
= (5/Mz)v,
(4.1)
In equation (4.1), a is the lattice constant. The kinetic energy per unit area o f wall is
therefore,
(4.2)
where M is the reduced mass of the titanium ion. The effective mass per unit area of wall,
p, is
p = A/5 2 / N a4.
(4.3)
The equation o f motion of the wall may be written as
p(d 2x / d t 2) + r(d x /d t) + qx = 2 PSE ,
(4.4)
where r represents damping effects, q the restoring force, associated, for example, with
local trapping of the wall, and Ps the saturation polarization. Damping may occur through
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coupling with lattice vibrations, selective impurity diffusion, local trapping, and acoustic
radiation, however, for convenience, r = 0 was assumed. The restoring force was defined
by,
(4.5)
q = 4PjL/ D X0en
Where
is that part of the static dielectric susceptibility arising from domain wall
displacements, and D is the domain width. In an alternating field,
= 4 £/D
'
q —pco2
0
such that resonance occurs at
(4.7)
Kittel set 5 = 0 .2 A, a = 4A, yielding p = 1 .0 x l 0 10/ N (g/cm2), and assuming N = I, and
taking Ps = 50,000 esu (16.6 (i.C/cm2),
~ 100, D * 10' 2 cm, the resonance frequency
due to 180° domain wall motion in B aT i0 3 was determined to be / 0 = 2 GHz. The absence
of resonance character in the relaxation spectra was accounted for by a combination o f
frictional effects and spread in domain widths.
The frequency dependence of ferroelectricity, including the apparent disappearance of
the ferroelectric response in the microwave region was considered by von Hippel (von
Hippel, 1952). In a ferroelectric material, such as B aT i03, permanent electric dipole
moments are constructional elements o f the crystal structure and are therefore, in general,
firmly anchored in place, and not available for free rotation. High frequency relaxation was
attributed to the change of the permanent net moment and with it, the creation o f a
mechanical deformation that travels with the velocity of sound. Since individual grains
have dimensions on the order of
1 0 '3
cm, and since the velocity of sound is about
cm/sec, resonance frequencies in the 100 MHz range are expected.
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1 0 5
100
The effect of piezoelectric grain resonance in ceramics was uniquely addressed by
studying the dielectric response of conventionally sintered and hot pressed lithium niobate
(L iN b03) ceramics, up to 1 GHz (Xi et al., 1983). Because L iN b0 3 possesses only 180°
domains; that is, there are no 90° domains, the frequency o f domain resonance can be
related directly to grain size. The high-frequency dielectric spectra of LiN b0 3 was shown
to have a distinct resonant quality, i.e., the dielectric constant, K, increases slightly, then
drops and saturates out to its clamped value, whereas tanS passes through a maximum.
Using an equivalent circuit model of grain resonance, the dielectric spectrum o f L iN b 0 3
was predicted. In the model, a series branch (L ,, C,, and R,) represents the mechanical
damping of vibration and a parallel branch, C0, represents the clamped high-frequency
capacitance. The value of the above parameterswere related to grain size, grain orientation,
and material constants, in a first approximation, byassuming a cubic grain, such that,
C = K sQEQd
(4.8)
C, =(ATr - K ‘ )e„d
(4.9)
(sdSe
^
( P - ^ ) e 0
(410)
and
D JpS1
-r* )e 0 •
<4 1 1 >
The grain size is given by d, Kr and Xs are dielectric constant under constant stress and
constant strain, respectively, SEis the mechanical compliance at constant field, p is the
density, £ 0 is the permittivity of free space, and D is the damping factor. The resonant
frequency, f Q, o f a grain is given by equation (4.12):
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101
/o
2
7
(4.12)
'
A number of piezoelectric equivalent circuits were connected together to simulate the
coupled resonant behavior of grains o f different size and orientation in a ceramic. Circuit
simulation software was used to determine the complex impedance of the network, from
which dielectric parameters were obtained. The calculated spectra was in excellent
agreement with the measured spectra.
Arlt (Arlt, et al., 1993, 94) proposed the existence o f strong Debye-like relaxation in
ferroelectrics due to the emission o f gigahertz shear waves from ferroelastic domain walls.
To illustrate the development of this theory, consider the hypothetical free paraelectric
crystallite shown in figure 4 .1(a).
(a)
(b)
h
d
H
d
Figure 4.1. Square face of cubic volume representing paraelectric crystallite (a), and
deformation of cubic crystallite due to development of P 0 (b).
Upon transformation from the paraelectric to ferroelectric phase, the cube of side d is
deformed due to the development of spontaneous polarization P0. Figure 4.1(b) shows the
deformed cube having a pair of domains separated by a 90° domain wall. In the presence
o f a static electric field, £ ,, and stress field, T I 3 = Ts, the domain wall shifts by Al in the x 3
direction, as indicated in figure 4.2. Consequently, mass is displaced by u x in the x x
direction in the upper domain, whereas mass is displaced by u2 in the -xx direction in the
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102
lower domain. The transformation also results in spontaneous strain, SQ, where for free
deformation, a = SQ.
The displacement, A/, in the simplest model, is governed by A ik = force per unit area,
where k is the spring or force constant per unit domain wall area. The term k contains
several contributions as indicated by equation (4.13).
k = km+ ke + kdef
(4.13)
These contributions include an elastic term due to the elastic clamping by the surrounding
media km, an electric term ke due to the electric field caused by the polarization charge at the
right and left boundaries of the displaced region, and kdef due to domain wall pinning by
defects. However, the electric term ke is implicitly considered through its association with
the electric field induced by the displacement o f the domain wall, and k ^ is assumed zero.
Consequently, instead o f (4.13), k = km is used.
d
Figure 4.2. Spontaneously deformed crystallite in the presence of a static electric field, E,
and stress field,Ts.
In the calculation o f the force per unit domain wall area, the electric dipole moment per
unit domain wall area, Ap, and the elastic dipole moment per unit domain wall area, Av,
caused by the displaced wall must be considered. Each of these moments have only one
non zero component (Arlt, et al., 1980):
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i
103
A/?, = 4 lP QM
(4.14)
and
Av5 = 2S0A1.
(4.15)
The energies per unit domain wall area in the presence of the electric field is, and stress
field Ts
are
W 'U c n c
= -V 2P0A/£,
(4.16)
and
^,asnc= -2SQAlT5.
(4.17)
The total energy is then given by
w,OIa/ = -V 2P 0 A/£, + (-2 S 0AlT5) = (Alk)Al.
(4.18)
By simply changing sign,
P0£, + 2SqT5.
AIk =
(4.19)
The change in polarization, A/*,, caused by the shift of the domain wall is
Ai> = V2P0 A //^ ,
(4.20)
and from geometrical considerations
«1 - o
2
= 2S0A l.
(4.21)
From (4.21), AI = —— — , and (4.19) becomes
2S0
2
-
« 2
=
—
S0 V2
7 ---- .
k
4SZTs
k
(4.22)
the first transducer equation. By equating Cm = 4502 /k and N = P0 /( V 2 S0), (4.22) can be
written as
u\ ~ u2 = NCmE\ + CmTs.
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(4.23)
104
The second transducer equation is arrived at by assuming distant electrodes such that AD,
caused by domain wall displacement equals AP x. The total dielectric displacement, D,, is
given by
D, = AD, + £„£,,
(4.24)
where e„is the intrinsic permittivity. From (4.20) and (4.21), (4.24) becomes
Dt = iV(«, —1*2 )/</ + £„,£,.
(4.25)
The transducing properties of the domain wall are within the ±AZ displacement region.
Provided the wavelength of the emitted acoustic wave is much larger than 2A/, the mass of
the domain wall can be neglected, and the domain wall is considered to be a local, infinitely
thin transducer. The static transducer equations become periodic when the static £, and Ts
fields are replaced with time varying fields having an exp(yG)f) dependence. The time
derivatives of (4.23) and (4.25), containing time varying fields, yield the dynamic
transducer equations:
= j(oCmNEl + j(oCmTs
(4.26)
D\d = N ( ux - u2 ) + jO E^dEi.
(4.27)
ti, and
The domain wall transducer can emit shear waves in the jc3 direction if the diameter of the
domain wall is much larger than the wavelength of the sound emitted. On either side of the
domain wall transducer, as well as in the material surrounding the domain pair, the
characteristic input impedance is given by Zm = -Jpc5s , where p is density and c5S is the
stiffness constant. The elastic shear stiffness constant of the grain in the rotated coordinate
system defined by figure 4.1 is given by c5S = —(c’,° + c33° —2c'n ° ), were c is the
4
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105
constant in the normal coordinate system. Since the domain wall transducer is coupled to
the input impedances, the relationship between stress and velocity is given by
(4.28)
T S = Z * “ 2 = - Z m« 2 .
Equation (4.28) can be used to eliminate
and Ts from equations (4.26) and
(4.27). The input admittance is given by D, / E\ . Integration of the input admittance yields
the total permittivity:
(4.29)
where
(4.30)
and
k
2
vS 5
(4.31)
In arriving at equations (4.30) and (4.31), km = 4 cssS q/ d , and the phase velocity,
v 55
= Vc 5 5 / P ’ have been used. The form of the elastic force constant, km, was arrived at
by assuming individual grains of a ceramic being embedded in an elastic matrix having the
same elastic constants as the grain. At frequencies below the acoustic resonance of the
individual grains, grains deform as much as adjacent grains will allow them to.
Consequently, according to their orientation, adjacent grains may either perfectly clamp
each other or may be excited into common vibration modes. However, at frequencies
above the acoustic resonance of the grains, domains still deform while preserving the gross
shape of the grain. The deformation caused by the domain walls is compensated by the
elastic shear deformation o f the domains, where the force constant derived from this elastic
deformation is km = 4 cssSqI d (Arlt, 1991). The relaxation frequency,/,, is related to the
relaxation time, x, through / , = 1/(2tcx) . Using (4.31), it follows that
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As previously stated, the domain wall emits shear waves when its thickness is much
larger than the wavelength of the emitted acoustic wave, however, when the domain wall
diameter is smaller than the wavelength of the emitted shear wave, Zmmust be replaced by a
complex impedance, which is strongly frequency dependent. Consequendy, for large
emitters (i.e., domain wall diameters > the sound wavelength), the relaxation is Debye­
like, as suggested by equation (4.29), however for small emitters (i.e., domain wall
diameters < the sound wavelength), the solution for etola, shows a decrease over a much
broader frequency range.
In summary it is expected that the high frequency relaxation in ferroelectrics should be
Debye-like, with a relaxation frequency dependent on the domain width, and relaxation
breadth dependent on domain wall diameter. However, mode coupling between
longitudinal and shear waves was not considered, and also, reversible elastic energy was
assumed to be stored exactly at the site of the domain wall, when in principle, the elastic
energy may be stored up to a depth d/2 above and below the domain wall. Consequently,
the dielectric response can moderately deviate from the expected Debye-like response.
Finally, the effect of temperature on AK can be predicted by substituting SQ = QPq into
(4.30), where Q is the electrostrictive constant. Because Q has a weak temperature
dependence, it becomes apparent that A K should be proportional to 1/P02. In the
ferroelectric state, P0 decreases with increasing temperature, therefore it is expected that AK
will increase with temperature up to the Curie temperature.
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107
4.2 High-Frequency Relaxation in Ferroelectric Ceramics
Powles (Powles, 1948) was one of the first investigators to characterize the highfrequency dispersion phenomenon in ceramic B aT i03. His work examined the highfrequency dielectric properties of a series of titanates, including magnesium, calcium,
strontium and barium titanate. Complex relative permittivity measurements were conducted
at 1.5 MHz, 9.45 GHz and 24 GHz. Through the use of quarter-wavelength (
matching slabs, and by measuring the power reflected and transmitted through a section o f
waveguide filled with the material under test, K ' and K " were determined. Dielectric
measurements revealed small changes in K ' for Mg, Ca, and Sr titanate between 1.5 MHz
and 9.45 GHz (the highest measured frequencies on those compositions), with an increase
in tan§ from ~ 10' 3 at 1.5 MHz to - 10' 2 at 9.45 GHz. B aTi03, however, exhibited
evidence o f significant relaxation with K ' decreasing from 1500 at 1.5 MHz to 300 at 9.45
GHz and finally to 126 at 24 GHz. Tan5 increased from 0.015 at 1.5 MHz to 0.53 at 9.45
GHz, reaching 0.59 at 24 GHz. Temperature dependent (20°C to 170°C) K '
measurements conducted at 1.5 MHz and 9.45 GHz exhibited dispersion both below and
above the Tc, with no observable change in Tc between the two frequencies.
Von Hippel (von Hippel, 1950, 1952) also reported the existence o f a relaxation
spectrum for ceramic BaTiG3. Although the details of the measurement techniques were
not provided, he reported the onset of relaxation at ~ 108 Hz, with a decrease in K ' from ~
1400 measured at 10 MHz to ~ 150 at 30 GHz. The decrease in K ’ with increasing
frequency seemed to be approaching a saturation value, i.e., the slope, d K '/d T
0,
however, as in the case with Powles, the corresponding measured loss values continued to
increase with increasing frequency. Direct current (dc) biasing effects were investigated by
applying a 18 kV/cm bias field. Measurements revealed a decrease in K ' from 1500 to
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I
108
1000 at 1 kHz, however with increasing frequency, K ' measurements conducted under a
bias field appeared to converge with those conducted under no bias field in the microwave
frequency region. Effects o f dc biasing on tan8 were not reported.
High frequency dielectric properties were also investigated in barium-strontium titanate
ceramics (Davis and Rubin, 1953). Using a filled coaxial line with matching
sections, temperature (-70°C to +100°C) and field dependent dielectric measurements were
conducted at 3 GHz by counting the number of Y i ^-resonances and measuring attenuation
in the (73% B aT i03-27% SrTi03) sample as temperature and field were varied. On
increasing the electric field, E, the maximum real relative permittivity (
)
decreased
from its zero field value of 5300 to 1500 under a 19.13 kV/cm applied field. Also, with
increasing E, the temperature corresponding to
, designated Tm, increased. A
comparison between low and high frequency (3 GHz) measurements revealed that the K '
measured above Tc showed virtually no frequency dependent dispersion; tanS measured
below Tc was markedly higher than that measured at low frequencies; and tan5 measured
above Tc rapidly decreased to low values.
Another study sought to measure the microwave dielectric properties of barium strontium titanate ceramics in the frequency range between 2.2 GHz and 3.5 GHz, using
two different methods (Iwayanagi, 1953). The first method involved measuring the
transmission resonances of a water filled cavity fitted with a moveable plunger to which
was affixed a slab of the dielectric material (Ba: 80%, Sr: 20%)TiO3. By adjusting the
plunger, with the attached dielectric slab, of thickness - A,g/4, A.g being the wavelength of
the propagating mode in the dielectric, variations with the first and second resonant points
were observed as the plunger was moved, thereby varying the resonant length, £ , o f the
cavity. By measuring the variation of £ and the power ratio at the second resonant point to
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109
the first, P ^P V and determining the propagation constant, y = a + *0 , in the dielectric,
a K ' —775 ± 15 and tanS = 0.14 ± 0.025 were determined at 3.08 GHz by a graphical
method. Similar measurements were carried out at temperatures between 30° and 80° C.
Compared to measurements conducted at 1 MHz, AT'was significantly lowered at 3.08
GHz with an associated increase in loss, however temperature dependent data at 3.08 GHz
exhibited a very broad peak at Tc. The author attributes this to the poor quality of the
sample.
In a second method the power transmitted through a filled rectangular waveguide (2.140
GHz - 3.530 GHz) at n X j2 transmission resonances was measured in B a-SrT i0 3 with 10,
20, 30, and 40% strontium titanate. This method allowed only for the determination of
K '. Again K ' was observed to decrease from its low frequency value. The measured K '
at 2.5 GHz was normalized to its 1 MHz value and plotted as a function o f percent S rT i0 3.
The
K '( 2 5GHz)
) value was extrapolated to 0% S rT i0 3, predicting a drop in K ’ for pure
B aT i0 3 to 0.7 of its low frequency value by 2.5 GHz.
Schmitt (Schmitt, 1957) conducted permittivity measurements between room
temperature and 160° C on ceramic B aT i0 3 of various densities at 9.4 GHz using a filled
waveguide technique. Measured densities varied between 4.5 and 5.4 g/cm 2. High density
BaTiOs samples yielded a K ' = 1060 and tan5 = 0.02 at 2 kHz, and a K ' = 560 and tan5 =
0.22 at 9.4 GHz. The ratio of high-frequency (9.4 GHz) to low-frequency (2 kHz)
permittivity ( ^ (9A G H z) was nearj ^
K\2.Q kH z)
same for ^
different samples, ~ 0 .5 , and did
not vary through temperatures up to the Curie point. Above the Curie point, the ratio
♦
gradually approached unity by 160° C, suggesting the persistence of dispersive effects
above the Curie temperature. The application o f a high dc bias field caused the high-
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110
frequency permittivity to change very little compared with the pronounced decrease
observed in the low-frequency value.
Petrov (Petrov, 1960) measured the dielectric properties of ceramic B aT i0 3 at 1 MHz
and 3 GHz. For the 3 GHz measurements he used a coaxial line terminated by the sample
and calculated the input impedance from the measured displacement of the minimum and
voltage standing wave ratio (VSWR). From the input impedance, dielectric properties were
determined, with a reported precision o f 5-10 % in K '. K ' and tan§ were measured under
bias fields from zero up to 17 kV/cm. The K ' = 1250 measured at 1 MHz decreased to
850, measured at 3 GHz under no bias field. The K ' = 850 and tan5 ~ 0.23 measured at 3
GHz under zero field, decreased to 500 and -0 .1 7 , respectively, under 17 kV/cm bias
field. K ' vs E, and tan 8 vs E loops retained their nonlinear behavior at 3 GHz. In
comparison to poly-domain single crystal BaTiOj, measured at the same frequencies using
the same techniques, the K ' o f the crystal remained higher with tanS remaining lower than
the ceramic, from 0 to 17 kV/cm applied bias field.
Gerson and Peterson (Gerson and Peterson, 1963) measured the high frequency
properties of lead zirconate titanate solid solutions with strontium titanate. To measure the
dielectric properties from 0.4 to 2 GHz, a coaxial line was terminated by the sample
capacitance. Calibration was achieved by replacing the sample with a short (metallic
dummy) of the same dimensions. This compensated for the impedance of the sample
holder arrangement, which then was treated as an inductor in series with the sample. The
impedance of the sample and its holder was determined by measuring the voltage standing
wave ratio (VSWR) and shift in position o f the voltage minimum. Dielectric properties
were determined from the measured impedance with a reported accuracy in K o f ~ 5% and
an uncertainty in tanS of 0.02. Room temperature measurements were conducted on
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Ill
paraelectric Pb0 3 5 Sr0 6 5 TiO 3 (Tc = 15 °C), and ferroelectric Pb0 9 4 Sr0 0 6 Zr 0 5 3 Ti0 4 7 O3.
Dielectric measurements conducted on paraelectric Pb 0 3 5 Sr0 6 5 TiO 3 showed no clear
evidence o f relaxation. The low frequency K o f 1600 remained virtually unchanged up to 2
GHz, with perhaps a slight decrease at the highest measured frequency. This decrease,
however, appeared within experimental error. TanS also remained relatively low and flat
(< 0.05) up to 2 GHz. Similar measurements conducted on ferroelectric
Pb 0 9 4 Sr0 0 6 Zr0 5 3 Ti 0 4 7 0 3 revealed clear relaxation with a decrease in K from its low
frequency value o f 1200 to its high frequency value o f 800. From the dispersion in K ,fr ~
1.4 GHz. Over the measured frequency range tanS continued to increase to > 0.4 at 2
GHz. A sample was then poled, yielding a d33 constant of 230 pC/N. The low frequency
K of the poled sample, approximately 1000, saturated out to the same value as that o f the
unpoled sample, approximately 800, at 2 GHz. Again from the observed dispersion in AT,
a relaxation f r was placed at = 1.5 GHz. TanS was lower than that measured for the
unpoled sample, also increasing with frequency and saturating out to 0.2. The expected
loss peak associated with classical relaxation phenomena was not apparent. These results
were attributed to the mechanism of piezoelectric domain resonance as suggested by
Devonshire (Devonshire, 1951); i.e., the relaxation was due to the resonance o f individual
domains and coupled groups of domains. From this point of view, the effect of poling was
to transfer part of the high-frequency relaxation o f the unpoled ceramic to some low
frequency resonance. In addressing Kittel’s theory (Kittel, 1951), it was acknowledged
that 180° wall motion (under low fields) is accompanied by no dimensional change, so that
piezoelectric activity and compliance are affected only by 90° domain wall motion. When
the 90° wall is set in motion, the above quantities are increased for the entire domain.
Consequently effects due to individual domains and those due to 90° domain walls can not
be separated, and should have the same upper frequency limits.
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112
Poplavko reported on the microwave dielectric dispersion o f various ferroelectric
ceramics and single crystals, including BaTiOa (Poplavko, 1964,
6 6
, 69). Recognizing
that the body of data accumulated on the microwave dielectric properties of B aT i0 3
consisted mainly of measurements conducted at discrete frequencies, Poplavko sought to
carry out dielectric measurements over a wide range (50 Hz to 16 GHz) o f frequencies, to
elucidate details of the relaxation spectrum (Poplavko, 1964). Because of the wide
frequency band of measurements, different techniques were employed to measure the
dielectric properties. Microwave measurements were conducted either by partially filling
coaxial lines (150 MHz to 3 GHz) with the test specimen or by use o f filled rectangular
waveguides (8.5-9.8 GHz and 12.5-16 GHz). The lower band waveguide (8.5—9.8 GHz)
reportedly yielded an error o f 3% in K and 30% in tan5. The higher band waveguide
(12.5—16 GHz) reportedly yielded an error o f 10% in K and 10% in tanS. Dielectric
measurements were carried out on poly-domain single crystal BaTi0 3 (discussed in the next
section), polycrystalline ceramic B aTi03, and ceramic Ba(Ti 0 gsZr0 USnO0 4 )O3. Through
the use of these different measurement techniques over different frequency regimes, the
data was connected together yielding continuous frequency dependent dielectric spectra.
Both ceramic BaTi0 3 and Ba(Ti 0 8 5 Zr0 IISn0 0 4 )O 3 exhibited a pronounced decrease in K
and increase in tanS, indicative of the onset of dielectric relaxation, in the frequency range
of 10 MHz to 10 GHz. Low field measurements conducted on BaTiOs showed K
decreasing from - 1500 at 1 MHz to ~ 600 at 10 GHz; tan 8 increased from ~ 0.02 to - 0.5.
Although these measurements showed neither evidence o f the high frequency saturation of
K nor decreasing tanS, Poplavko placed the center of the dispersion between 7-8 GHz.
The application of a strong bias field seemed to have no effect on the center frequency of
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113
the relaxation, however, both K and tanS were lowered, with K coming into coincidence
with the unbiased K at the highest measured frequencies.
By measuring XIA resonances o f the reflected signal from polished specimens fitted in a
waveguide, Poplavko determined the dielectric properties of a series of ferroelectrics at
selected frequencies between 10 GHz and 50 GHz (Poplavko, 1966). The reported error
in K from these measurements was ~ 3%. Measurements were conducted both below and
above the Curie temperature on stoichiometric and doped B aTi03, as well as several other
ferroelectric and antiferroelectric compositions. In this work, microwave dielectric
dispersion was reportedly observed in ferroelectric B a T i0 3, but not in the paraelectric
phase, i.e., the permittivity above the Curie temperature at microwave frequencies agreed
with that measured at low frequencies. However, for some BaTi0 3 solid solutions
(BaogsCao ojTiOj and Ba Ti0 8 5 Zr 0 1 5 0 3), microwave dielectric dispersion was observed 50
to 100 °C above the ferroelectric to paraelectric phase transition. It was suggested that the
presence o f the dopants in BaTi0 3 led to dielectric dispersion in the paraelectric phase.
Finally, by using a variant of the waveguide resonance technique whereby the
rectangular waveguide was replaced by an air strip line, Poplavko extended the
measurement to millimeter wavelengths (0.3 to 80 GHz), and improved the accuracy of
both the measured K and tan 8 values (Poplavko, 1969). Figure 4.3 shows the measured
results. The waveguide measurements of K and tanS conducted on ceramic B aT i0 3,
through 40 GHz, showed that the microwave dispersion ends at millimeter wavelengths in
ferroelectric B aTi03, with the absence of microwave dielectric dispersion in the paraelectric
phase. The measured relaxation exhibited a decrease in K from ~ 1000 to 500, with a peak
tanS = 0.5-0.6, measured at 4.2 GHz. The value o f K changed very little between 10 and
80 GHz. Ceramics prepared from various technical grades of BaTi03, showed markedly
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114
different room temperature Ks (1000-6000) and A ^ s (7000-11000), however, at
frequencies between 20-80 GHz, K= 500 ± 50 and tanS = 0.15 ± 0.05 for all samples,
suggesting the microwave relaxation was due to the polydomain structure o f the
ferroelectric ceramics. Above the Curie temperature no dispersion was observed up to 46
GHz. however, at 61 GHz and 75 GHz, frequency dependent dispersion was observed in
the paraelectric phase. The very high frequency dispersion occurring at millimeter
wavelengths in the paraelectric phase, was believed to have been related to the dynamic
theory of ferroelectricity, whereby the frequency of the “soft” mode of lattice vibrations
(transverse optical phonons), decreases with temperature.
r i ta n 5
I
mo
boo
ZOO - 0 2
Figure 4.3. Frequency dependence of dielectric constant (1), and dielectric loss (2)
(Poplavko e ta /., 1969).
More recently, dielectric measurements through the relaxation frequency were conducted
on stoichiometric ceramic B aT i03, its solid solutions containing either B-site Zr or Hf
cations, and its solid solutions containing A-site Ca cations (Kazaoui et al., 1991).
Measurements were conducted between 1 MHz and 1 GHz by measuring the reflection
coefficient of the sample placed at the end of shorted coaxial line. Although the
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measurements were limited to
1
GHz, with part o f the relaxation spectra truncated, several
compositionally dependent trends in relaxation f r e q u e n c y , w e r e observed. With
increasing Zr concentration in BaTi^jZ^Oj (x = 0, 0.1, and 0.2) ,/ r decreased from 500
MHz, to 300 MHz, to 90 MHz. Increasing Hf concentration yielded a similar result
whereby / r decreased from 500 MHz to 300 MHz to 50 MHz. Calcium additions resulted
in the opposite effect, that is, f Tincreased with the increased calcium admixture.
Compositions of the form Ba,.vCavT i0 3 (y = 0,0 .0 5 , and 0.10) showed an increase o f /r
from 500 MHz to 700 GHz, to > 1 GHz. Temperature dependent measurements were
carried out through the Curie temperature of BaTi0 gZr0 2 0 3 (~ 35° C) as well as through the
tetragonal to orthorhombic phase transition of B aT i03. These measurements revealed a
minimum in /r at both the Curie temperature (in BaTi^ZTg^Oj) and lower tetragonal to
orthorhombic phase transition (in B aTi03). In B aT i03, the temperature dependence o f /r
was otherwise flat. Figure 4.4 illustrates the temperature dependence o ff r measured in
BaTiOv Microstructure analysis showed that both Zr and H f additions reduced average
grain size to ~ 15 pm, whereas Ca additions increased grain size to ~ 30 pm.
fr(H«l
I09
•7
S
4
z
1
*S7
4
— j~~
MO
]
100
1
!
j
!
;
!
:
i
i
•
I
1
11
1
- 1 !
r n
;
!
:
JL
107------ — ---------------- V
■■
... |
1
i
i
i
}
’
!
T
1 J0
4 0 0 TiK
Figure 4.4. Relaxation frequency versus temperature for B aT i0 3 (Kazaoui et al., 1991).
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116
Microwave dispersion in lead zirconate titanate (PZT) ceramics o f the composition
Pb(Zr,_xTix) 0 3 were investigated as well (Bottger and Arlt, 1992). PZT is also a perovskite
ferroelectric whose Curie temperature is almost twice that of barium titanate and possesses
almost twice the ferroelectric lattice distortion and polarization. Dielectric properties were
measured from 3 MHz to
6
GHz by measuring the reflection coefficient of a 50Q coaxial
line terminated by the sample. In this case the sample radius was kept less than half the
radius of the inner conductor, and the measurement line was calibrated using three arbitrary
standards o f known dielectric properties with dimensions identical to the test specimen. On
decreasing x from 0.48 to 0.42, low frequency K decreased,^ decreased from 1.2 GHz to
400 MHz, and the relaxation broadened, with the high-frequency K lowered by
approximately half its low frequency value. The relaxation behavior of Pb(Zr0 5 2 Ti0 4 g)O 3
was studied as a function of temperature. With increasing temperature, low frequency K
increased, the relaxation step, AK increased, and peak tanS increased, however, f r
remained unchanged. This, along with observations made by Kazaoui, suggest that the
relaxation is not a thermally activated process.
Using the measurement techniques just described, high frequency dielectric
measurements were conducted on B aT i0 3 ceramics and reported along with the PZT
results just discussed (Arlt, et al., 1994). The objectives of this work were to determine
temperature, grain size (g.s.), and internal bias field (via Ni-doping) effects on the highfrequency relaxation in both PZT and B aT i03. Using the relaxation model proposed by
Arlt (Arlt, 1993,94), and presented in section 3.1, relaxation curves were calculated for
B aT i0 3, using P0 = 0.26 C/m2, SQ= 0.01, c 5 5 = 34.02 GN/m 2 and p = 6.02xl0 3 kg/m3.
The domain width was assumed to be d * 1 pm and the intrinsic dielectric constant,
~ 1100. The measured and calculated relaxation spectra were compared, revealing
good qualitative agreement.
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i
117
The temperature dependent PZT results, previously discussed, were examined within
the context of the proposed theory. The increase in AK with increasing temperature (below
the Curie temperature) was explained by the decrease in PQwith increasing temperature,
where AAT was shown to be «= l/P^ . Coarse grain BaTi0 3 yielded sim ilar results, i.e., AAT
increased as temperature increased (P0 decreased).
Aged B aTi0 3 doped with up to 0.2 mole % Ni was measured. Ni—doping introduced
internal bias fields which had the effect of increasing relaxation frequencies and lowering
the relaxation steps, AAT.
Finally, measurements on coarse grain and fine grain PZT and B aT i0 3 were carried out.
The coarse grain PZT (g.s. ~ 20 (im) showed expected Debye—like relaxation with f r ~
200-300 MHz, AT' decreasing from 600 to 400, and K" peaking to 100. The fine grain
PZT (g.s. ~ 1 fim) also showed relaxation with AT' decreasing from 650 to 400 and AT"
peaking to 300 at f r ~ 1 GHz. The coarse grain B aTi0 3 (g.s. ~ 50 |im ) showed typical
relaxation w ith /r ~ 700 MHz. K ' decreased from ~ 1600 and appeared to saturate out to ~
900. Dielectric measurements conducted on the fine grain B aTi0 3 (g.s. ~ 0.2 fim) yielded
a broader relaxation. AT' decreased from ~ 3900, approaching no apparent saturation value
over the measured frequency range; at ~ 4 GHz, AT' ~ 1500. K " was also much higher
compared to the coarse grain B aT i03, over the entire frequency range o f measurement. A
broad peak in K " was centered at ~ 1 GHz. Arlt suggests that for a g.s. ~ 0.2 fim, the
emitting domain walls have diameters much smaller than the wavelength o f the sound
emitted, thus explaining the broad relaxation curve.
In summary, the literature has consistently supported the existence o f high-frequency
(microwave) relaxation in ferroelectrics below the Tc. Table 4.1 summarizes the dielectric
properties of ceramic B aT i0 3 at frequencies both below and above the frequency range o f
relaxation. The discrepancies between the various reported values are likely due to
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118
variations in sample purity, density, microstructure, and the accuracy and precision
associated with different measurement techniques.
Table 4.1 Summary of Reported Dielectric Properties of Ceramic BaTiOj Below and
Above the Relaxation Frequency
1 MHz
^
1500
tanS
0.015
10 GHz
K
tanS
300
0.53
1400
0.01
200
0.7
1060
0 . 0 2
560
0.22
1500
0.02
500
0.5
AK
peak tanS
f-
550
0.6
4.2 GHz
800
0.4
0.7 GHz
reference
Powles,
1948
von Hippel,
1950
Schmitt,
1957
Poplavko,
1964
Poplavko,
1969
Arlt, 1994
In general, the relaxation frequency has been placed between 0.5 and 10 GHz, with K
decreasing to 0.1 - 0.5 of its low frequency value. The dielectric step, AK, has been
shown to increase with increasing temperature (Bottger and Arlt, 1992), whereas / appears
temperature independent, except at temperatures near or through phase transitions (Kazaoui
et al.\ Bottger and Arlt, 1992). A reduction in grain size apparently had the effect of
increasing/,. (Arlt, et al., 1994). In addition, poling and high bias fields lowered K and
tanS (Hippel, 1950; Davis and Rubin, 1953; and Gerson and Peterson, 1963); poling has
been shown to s h i f t / to slightly higher frequencies (Gerson and Peterson, 1963). Some
observations o f the high-frequency relaxation, however, remain in conflict. Some have
reported frequency dependent dispersion both below and above the Curie temperature
(Powles, 1948; Schmitt, 1957; and Poplavko, 1966), while others have reported the
•< :
__
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119
absence of dispersion above the Curie temperature (Davis and Rubin, 1953; and Poplavko
1966, 1969).
4.3 High-Frequency Relaxation in Poly-Domain Single Crystal BaTiOj
Paralleling investigations conducted on ceramics, were studies conducted on BaTiOj
single crystals. This section reports observations made on polydomain single crystals of
B aT i03.
In an attempt to discern whether the relaxation phenomena was due to the ceramic form,
or instead, an intrinsic property of ferroelectric BaTi03, Fousek (Fousek, 1958) measured
the room temperature dielectric properties o f polydomain single crystal B aTi0 3 at 1 GHz.
Although the details of the measurement technique were not provided, K ' of five different
samples was measured at 1 kHz and 1 GHz, under zero and up to 7 kV/cm bias field. K '
measured under no bias at 1 kHz ranged between 2230 and 4080, whereas at 1 GHz K '
K'(lG H z)
ranged between 1620 and 3180. The ratio, ------------- , varied between 0.69 and 0.77,
K '(lkH z)
comparable to ceramics at similar frequencies. By applying a bias field up to 7 kV/cm, the
K ' vs E(bias field) loops measured at 1 GHz retained their nonlinear character.
Petrov (Petrov, 1960) measured the dielectric properties o f polydomain single crystal
BaTi0 3 at 1 MHz and 3 GHz. For the 3 GHz measurements he used a coaxial line
terminated by the sample and calculated the input impedance from the measured
displacement of the minimum and VSWR. From the input impedance, dielectric properties
were determined. K ' and tanS were measured under bias fields from zero up to 17 kV/cm.
The K ' = 2700 measured at I MHz decreased to 1800, measured at 3 GHz under no bias
field. The K ' = 1800 and tanS ~ 0.14 measured at 3 GHz under zero field, decreased to
950 and ~ 0.1 1 , respectively, under 17 kV/cm bias field. K ' vs E, and tanS vs E loops
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retained their non-linear behavior at 3 GHz. In comparison to ceramic B aT i0 3, measured
at the same frequencies using the same techniques, the K ' of the crystal remained higher
with tanS remaining lower than that of the ceramic, from 0 to 17 kV/cm applied bias field.
Using a cavity perturbation method, the dielectric properties o f poly-dom ain single
crystal B aTi0 3 were measured at 3.3 GHz from room temperature through the Curie
temperature (Nakamura and Furuichi, 1960). K ' measured at room temperature was ~ 500,
decreased slightly with increasing temperature, and peaked to 6000 at Tc. Above Tc, K '
decreased in accordance with typical Curie—W eiss behavior. Below Tc, K ' was
significantly lower than that measured in single-domain single crystals (discussed in the
next section) at 24 GHz. Above Tc, K ' was in agreement with that measured in single­
domain single crystals. In addition, below Tc, tanS was too high to measure, however
above Tc, tanS rapidly dropped to ~ 0 .0 1 , significantly lower than that reported (tanS -
0
.1 )
in single-domain single crystals. The large dispersion observed below Tc was attributed to
a mechanism connected with the poly-domain state o f the crystal.
4.4 High-Frequency Relaxation in Single-Domain and Paraelectric Single Crystal B aTi0 3
The measurement of dielectric parameters of single-domain single crystal B aT i0 3 was
conducted at 24 GHz as a function of temperature (Benedict and Durand, 1958). Crystals
polarized in the c direction and approximately 300 pm thick, were used to fill the cross
section of a /((-band waveguide flange. From measurements o f the phase shift and
attenuation, the dielectric constant and loss tangent of the sample were obtained.
Temperature dependent measurements through the Curie temperature revealed typical
Curie—Weiss behavior. The value of tanS was observed to be somewhat lower than that
observed for B aTi0 3 ceramics at microwave frequencies. TanS remained nearly constant at
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121
~ 0.1 except in the vicinity of the ferro- to paraelectric phase transition. The dielectric
constant in the ferroelectric regime was about one—half that observed at low frequencies,
sharply increased at the Curie temperature to ~ 6000, and was in essential agreement with
low frequency values above the Curie point. The decreased K measured in the ferroelectric
regime was attributed to the clamping o f the mechanical vibrations above the piezoelectric
resonance o f the sample, and not to a loss o f ferroelectricty.
Dielectric measurements were conducted on single crystal BaTiOs in the paraelectric
region, from 160 °C down to the Curie temperature, at X—band (8.2 - 12.4 GHz)
frequencies (Lurio and Stem, 1960). The technique involved measuring A/2 transmission
resonances in a filled waveguide flange. A number of crystals of different thickness were
used to measure K ' through the paraelectric region. The temperature dependence o f K '
from 160° C through Tc agreed well with their low frequency data. The expected C urieWeiss behavior was observed, with a Curie constant = 3.77 x 10' 5 °C '. In general, no
clear frequency dependent dispersion was observed in the paraelectric region.
Ballantyne (Ballantyne, 1964) measured the reflectivity and transmission of singledomain single crystal BaTiOj, polarized along the c-axis. Measurements were conducted
through microwave, millimeter, sub-millimeter, and infrared frequencies (
1
to
1 0 0 0
cm-
1), at discrete temperatures from room temperature through Tc. Values of K ' and K " were
calculated by means of a Kramers—Kronig analysis and the Fresnel equations for normal
incidence. The main features of the curves are a strong temperature dependent, rapidly
decaying maxima in K \ below
2 0
cm '1, and resonances due to lattice vibrations at 182 and
491 cm '1. The K ' maxima decreased with increasing temperature, from 2000 measured at
24 °C to 1500 measured at 100 °C, sharply increased to 5000 at 127 °C, above which, it
began to decrease again.
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122
Measurements o f K ' and K " at 24 GHz were conducted by measuring the VSWR in
front o f a thin crystal plate (31 pm) spaced a A/4 from the shorted end of a waveguide. The
complex permittivity was in close agreement with that reported previously (Benedict and
Durand, 1958), with the measured K ' below Tc equal to about one—half the low frequency
value. In addition, reflectivities calculated from measured K ' and K " agreed to within ±
1
% of those measured.
In summary, microwave dielectric dispersion has been reported for poly—domain single
crystal ferroelectric B aTi0 3 below Tc, and is apparently absent in single-domain single
crystal B aT i0 3 below Tc. In single-domain single crystal B aTi03, the decreased dielectric
constant below Tc at microwave frequencies was explained by the piezoelectric clamping of
the macroscopic sample. The absence o f dielectric dispersion in the paraelectric phase up
through millimeter frequencies has been consistently reported. This along with the
previously reported observations on ceramic ferroelectrics, clearly suggests that the
microwave dielectric relaxation is intimately connected with the domain structure, and is
likely due to the individual and coupled vibrations of grains and/or domains.
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CHAPTER V
OBJECTIVE
Thus far, it has been shown in the previous chapters, that polydomain ferroelectric
BaTi0 3 ceramics and single crystals exhibit high frequency Debye-like relaxation. The
apparent absence o f this relaxation in single domain single crystals clearly illustrates that the
relaxation phenomena is very likely associated with the domain state of the ferroelectric. If
indeed the domain structure governs the high frequency properties of these materials, the
potential for a novel approach in the high frequency tunability o f the dielectric properties of
these materials exists. Specifically, it should be possible to tune the high frequency
properties o f these materials by shifting the relaxation frequency through controlling the
domain state. It was demonstrated in chapter II that the domain structure of ferroelectrics is
highly dependent upon grain/particle size. Although the boundary conditions o f a grain in a
ceramic are dramatically different from those experienced by a free particle, it was shown
that both grains and free particles are expected to undergo several domain state
transformations, generally progressing from a polydomain state to a single domain state
with decreasing size. The objective o f this work, therefore, is to investigate the effects of
this domain modulation with decreasing grain/particle size on the high frequency properties
of a prototypical ferroelectric.
The ferroelectric composition selected for this study is BaTiOj. Barium titanate is a well
understood ferroelectric compound, widely used in the electroceramics industry, and has
been the subject of much research since its discovery. As previously discussed, B aT i0 3 is
a well behaved, first order prototype ferroelectric. It is for this reason that B aT i0 3 was
selected, however, it is anticipated that the results of this study are applicable to other
ferroelectric compositions.
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124
This research attempts to examine the high frequency properties o f B aT i0 3
polycrystalline ceramics as a function of grain size (10 to 0.2 (im), and B a T i0 3 particles as
a function o f particle size (1 pm to 30 nm). The B aT i0 3 particles will be assembled as 0-3
composites using a low K, low loss polymer matrix. The selected matrix material should
possess frequency independent dielectric properties through the frequency range of
measurement. The measured change in the dielectric properties of the composite with
frequency may then be attributed to the ferroelectric phase. In accomplishing the above
objective, various microwave measurement techniques will be employed, and thus their
capability to yield reliable reproducible results will be evaluated. This will be realized by
fabricating and measuring the microwave dielectric properties of multiple samples
whenever possible. This should reveal the error and precision associated with each of the
various measurement techniques.
The investigation of ceramics will serve to:
•
Establish the feasibility of the selected high frequency measurement techniques. As
evident in chapter IV, microwave measurements o f ferroelectrics present a formidable
challenge. Measurements conducted on conventionally prepared B aT i0 3 ceramics may
then be compared to those in literature. This will help to validate the selected
measurement techniques.
•
Examine the effects of domain modulation on the high frequency properties of B aT i0 3
ceramics. The domain structures o f ceramics and free particles are expected to have
different size dependent configurations; the consequences of their respective size
reductions should yield different high frequency properties. In the case o f the ceramic,
chapter II showed how domain patterns in ceramic grains undergo more complicated
twinning due to stress relief.
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125
The investigation of particles will serve to:
•
Observe the relaxation phenomenon in a less complicated system.
1. It was shown that in the ceramic, domain and grain resonances were likely
coupled to adjacent domains and grains. For this reason, the interpretation o f
the relaxation phenomena is inherently more difficult in ceramics. However, in
the case of free particles, these grain resonances are decoupled; that is, the free
particles are not coupled to surrounding particles.
2. In addition, particles assembled in a polymer matrix should experience
significantly less stress due to their surroundings, thus, the twinning associated
with stress relief should be reduced, leading to a domain state intrinsic to
particle size and not due to stress relief.
•
Establish a better correlation between relaxation frequency and intrinsic domain state.
By using the relaxation models introduced in chapter IV, domain size may be calculated
from measured relaxation frequencies and correlated with particle size.
Finally, this work also attempts to correlate microstructural observations, conducted as
part of this work as well as reported in literature, with the measured relaxation spectra.
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CHAPTER VI
EXPERIMENTAL PROCEDURE
In this chapter, processing and sample preparation, as well as measurement equipment
and techniques are discussed. First, the selection and characterization o f the commercial
powders are considered. Next, the processing of ceramic and composite samples as well
as their microstructural characterization are examined. Then, sample preparation and low
frequency ferroelectric and dielectric property measurements are discussed. Finally, the
high frequency dielectric measurement techniques employed in this work are discussed.
6
.1 Starting Powders
The ceramic and composite materials used in this study were prepared from several
sources of commercial powders. Table 6 .1 shows the various sources of powders and
summarizes how they were employed in this study.
Table 6 .1 Commercial Powder Sources
Powder
Comments
Use
Manufacturer
Barium Titanate,
code 219-9
mixed oxide
coarse grain
ceramics
Ferro
Corporation/T ranselco
Division
BTOl
Hydrothermal
(p.s. ~ 90 nm)
fine grain ceramics
(g.s. < 1 pm)
Sakai Chemical
Industry Co., Ltd.
TICON HPB
Chemically
precipitated (oxalate)
(p.s. - 1 pm)
composites
TAM ceramics, Inc.
BT - 8
Hydrothermal
(p.s. ~ 0 . 2 pm)
composites
Cabot Performance
Materials
BT-16
Hydrothermal
(p.s. ~ 30 nm)
composites
Cabot Performance
Materials
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127
6
.1.1 Powder Characterization
The powders used for composite fabrication were characterized in terms of their surface
areas, particle size distributions, X -ray diffraction patterns, and per cent weight loss on
heating. These techniques are briefly summarized here.
6
.1.1.1 Specific Surface Area and Particle Size Distribution
Surface areas of the powders used for composite fabrication were obtained by the single
point nitrogen adsorption technique (Brunaur, Emmett and Teller (BET) technique) using a
Quantachrome Monosorb Surface Area Analyzer1. Surface area values for each powder
were averaged from three measurements. The Monosorb analyzer determines the surface
area of a powder by measuring the quantity o f a gas adsorbed on a solid surface, which
was sensed by a change in the thermal conductivity of a flowing mixture o f an adsorbate
and an inert carrier gas, usually nitrogen and helium, respectively. The particle size o f the
powders was determined from surface area measurements by using the following equation
(Stockham and Fochtman, 1977):
D = —^— ,
P(£A)
( 6 . 1)
where D is the primary particle size (pm), p is the theoretical density (g/m3), and SA is the
specific surface area (m 2 /g).
Particle size distribution (PSD) of the powders used for composite fabrication were also
obtained. For the TICON HPB powder, particle size distribution was determined using a
sedimentation apparatus2. The method employs X -ray absorption to measure the fractional
mass of the powder sample that settles from a dispersion of the powder in a viscous liquid.
This fractional mass is related to its particle size. To obtain a proper dispersion and
Model MS—12, Quantachrome Corp.
Micromeritic Sedigraph Model 5000, Micromeritic Inc.
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iI
128
subsequent settling o f the particles during measurement, it was necessary to disperse the
powder in a suitable liquid medium. A homogeneous suspension o f pow der was achieved
using an ultrasonic bath and Sedisperse3, which is a highly purified, non—polar saturated
aliphatic hydrocarbon surfactant.
To analyze the PSD of the hydrothermal powders (BT- 8 and BT—16), a laser diffraction
technique was employed4. The only material qualification to use this technique is that the
refractive index of the material must be different from the medium in which it is dispersed.
Dispersed particles momentarily traversing a collimated light beam will cause Fraunhofer
diffraction of light outside the cross section o f the beam when the particles are larger than
the wavelength of the light. The intensity o f the forward diffracted light is proportional to
the particle size squared, but the diffraction angle is inversely proportional to the particle
size. A He-Ne laser is used to form the collimated and monochromatic beam o f light. The
combination of an optical filter, lens and photodetector interfaced to a computer, enabled
the computation of particle size distribution data.
A small amount of sample (< 0.2 g) was placed in a mixture o f 40 ml o f de-ionized
water and 0.5 g of a 30 weight per cent solution of Sodium DiMethylHexaphosphate. The
mixture was placed in an ultrasonic bath for 5 minutes to ensure complete dispersion. The
correct amount of the dispersion was then introduced into the laser sizer and a measurement
taken.
6
.1.1.2 Thermal Analysis
Thermal analysis was conducted on each o f the powders used for composite fabrication
to reveal the presence o f residual organics and absorbed hydroxyl groups (Noma, et al.,
1996; and Wada, et al., 1996). Thermogravimetric analysis (TGA ) 5 measurements were
3
4
5
Micromeritic Sedigraph Model A -l 1, Micromeritic Inc.
Microtrac Ultrafine Particle Analyzer
Perkin Elmer, DTA 1700
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129
performed by introducing small amounts ( ~ 40 to 50 mg) of each powder into the
instrument’s sample pan. Weight change with temperature was recorded from 25 to 1000°
C at heating rates between 10 and 20° C/min. TGA data on TICON HPB powder showed
no weight change up to 1000° C. Data collected on the hydrothermal powders, however,
showed a precipitous decrease of ~ 1.4 weight per cent between 220° and 410° C. Another
weight drop occurred between 770° and 850° C of less than 0.3 weight per cent.
Consequently, the hydrothermal powders used for composite fabrication were annealed
at 500° C in oxygen for three hours, prior to subsequent use. The annealing had no effect
on the particle size. Although the TICON HPB showed no weight loss associated with the
presence o f residual organics, this powder was annealed in oxygen at 1100° C for two
hours. This was done so as to remove any anomalous surface layers on the particles,
possibly introduced during any milling or grinding steps associated with its processing.
Such layers have been shown to decrease the room temperature tetragonality as well as lead
to remnant tetragonality above the Tc; annealing under the above conditions was shown to
increase room temperature tetragonality and eliminate remnant tetragonality above Tc
(Schoijet, 1964; and Goswami, 1969).
6
.1.1.3 X -R ay Diffraction
To determine the phase purity and lattice parameters of the powders used for composite
fabrication, X—ray diffraction measurements were performed using an automated
diffractometer employing Ni-filtered Cu K a radiation with a tube voltage and current o f 40
kV and 35 mA, respectively. The diffractometer6 scan ranged between 20-70° 20 at a rate
of 0.02° 20/min. The annealed powder samples were prepared by filling a small recessed
volume in a glass slide.
6
SCINTAG Automated Diffractometer
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130
6.2 Ceramic Processing
Coarse grain (10-20 fim) BaTi0 3 polycrystalline ceramics were processed using
conventional solid state sintering. Disks were prepared by mixing the powder (see Table
6.1) with 3—5 weight p ercen t polymer binder7. The powder-binder mixture was pressed
into I inch diameter pellets at pressures between 20,000 and 40,000 psi. This was
followed by binder burnout by heating to 425° C for 2 hours and then heating to 600° C for
1 hour. The disks were then sintered at 1350° C for 3 hours in a flowing oxygen
atmosphere.
Fine grain ceramics (g.s. ~
1
pm and g.s. ~ 0.2 pm ) were prepared using hydrothermal
starting powders (see Table 6.1). Fine grain ceramics of g.s. ~ 1 pm were prepared by dry
pressing the powder into 1 inch diameter pellets. The pellets were first uniaxially pressed
at ~ 9 MPa followed by cold isostatic pressing at ~ 280 MPa. The pellets were then hot
pressed at 53 MPa at 1150° C, for 30 minutes in an Argon atmosphere. The fine grain
ceramics of g.s. ~ 0
.2
pm were prepared in nearly an identical manner, however, the
starting powder was first acid-washed in dilute H N 0 3 acid (pH - 4.0). All fine grain
ceramic samples were annealed in oxygen at 1100° C for 30 minutes.
6.3 Composite Processing
Attempts were made to exploit various polymers for use as the matrix material in the
fabrication o f the composite samples. One approach involved mixing the BaTi0 3 powder
with a Spurrs 8 low-viscosity embedding medium. Powder loadings up to 40 volume %
were attempted. The powder-polymer mixtures were packed in phenolic or polyethylene
molds and cured over night at 70 °C. Unfortunately, the resulting composites where
characterized by excessive porosity and low densities. Another approach explored the use
7
8
Acryloid Resin, Rohm and Haas
Polysciences, Inc.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
131
of polytetrafluoroethylene9 as a matrix material. In this case, 750 fim
polytetrafluoroethylene powder was “cold” milled with B aT i03 powder. The milling
served to reduce the particle size of the polytetrafluoroethylene and aid in the mixing
between the powder and polymer. Chips o f dry ice (solid C 0 2), along with the BaTi03 and
polytetrafluoroethylene powders were added to a polyethylene bottle partially filled with
zirconia media, and closed. A small pin hole allowed for excess pressure to be released.
The powders were mixed and milled on a vibratory mill for ~ six hours. The mixture was
then removed from the mill and the dry ice was allowed to sublimate off. The BaTi03polymer mixture was pressed into disks at 370 °C. Composites fabricated this way using
the TAM powder were - 9 0 % dense. The composites fabricated this way using the
hydrothermal powders had significantly lower densities (< 80 %) and showed evidence of
reduction; consequently, this method was abandoned.
Ultimately, composite samples were prepared using an electrically heated mixing
preparation center10. Polypropylene11 was selected for the matrix material because o f its
thermal stability (Tm= 176° C) and reported low-loss, high frequency dielectric properties
(K = 2.256, tanS - 10"4) (Afsar, 1985). The mixing chamber used had a 60 ml capacity.
The amount of material put into the chamber necessary for optimum mixing was determined
by the simple formula: sample charge (g) = loading factor x specific gravity of composite
(g/cc). The loading factor depends on the sample chamber volume and mixing head type.
Taking the density of BaTi03 as 6.0 g/cc and that of polypropylene as 0.90 g/cc, the
expected density o f a composite material comprised of a particular volume fraction of
BaTi03 is straightforward. Using this composite density and a loading factor of 40,
optimum sample charge was calculated. The mixing chamber was heated to 180 °C, at
which time an appropriate amount of polypropylene pellets were added. A roller style
9
10
Polysciences, Inc
C. W. Brabender
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132
mixing blade designed for high-shear-rate mixing, necessary for thermoplastics was
selected. Mixing rate was set at 20 rpm. Once the polymer was completely melted, an
appropriate amount o f B aT i03 powder, depending on desired volume fraction of B aTi03,
was added to the mixing chamber. Mixing occurred for 30 minutes, at which time the
composite melt was removed from the chamber and cooled. The composite was then added
to heated (180° C) dies and pressed at pressures not exceeding 5000 psi. Efforts were
made to maintain constant pressure on all samples as they cooled.
6.4 Characterization of Ceramics and Composites using Scanning Electron Microscopy
(SEM)
The microstructure of sintered BaTi03 ceramic samples was examined from
photomicrographs o f either chemically etched or thermally etched surfaces. Cross sections
o f the coarse grain and fine grain ceramics of g.s. ~ 1 pm were polished to a mirror finish
using increasingly finer grades of SiC polishing powders: 20, 12, 5, and 3 pm. This was
followed by a finishing polish using diamond paste (1 and 0.5 pm) to provide a m irror-like
finish. These samples were then chemically etched using a 5 per cent by weight, dilute
solution of hydrochloric acid (HC1), to which a few drops of hydrofluoric acid (HF) were
added. The samples were immersed in the etchant from 30 to 120 seconds, while being
periodically pulsed with ultrasonic waves. The fine grain ceramic o f g.s. ~ 0.2 pm was
polished using the same techniques and thermally etched at 1100° C in oxygen for 30
minutes. Each sample was then affixed to a brass sample mount using double-sided
carbon tape, and gold-coated for two minutes to prevent charging in the SEM 12.
Microstructures were observed and photographed using the SEM.
Himont U.S.A., Inc.
Model IS I-D S 130, International Scientific Instruments, Inc.
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133
Composite samples were also examined using the SEM to reveal the connectivity o f the
ferroelectric phase. Using sim ilar techniques, except no etching of any sort was employed,
SEM microstructures were observed.
6.5 Characterization of Hydrothermal Powders Using Transmission Electron Microscopy
(TEM)
Transmission electron microscopy was used to investigate the particle morphology and
possible existence of any domain twinning. Samples were prepared by crushing the
powders in an agate mortar with isopropanole. A drop o f the dispersion was then
transferred to a TEM grid coated with an amorphous carbon film. Once the powders dried,
bright field and dark field TEM observations were made using a Philips 420 STEM13
operated at an accelerating potential of 120 kV.
6.6 Low Frequency Dielectric Property Measurements
Room temperature dielectric constant and dissipation factor were determined on all
ceramic and composite samples. Prior to measurement, all samples were ground parallel,
cleaned in an ultrasonic bath o f acetone, electroded with sputtered gold to which a small
amount o f air dry silver paste was also applied, and placed in a drying oven (150° C) for at
least 1 hour. Upon removal from the drying oven, the samples were allowed to equilibrate
to room temperature, at which time measurements were conducted.
The capacitance (C) and dissipation factor (tan8) o f the samples were measured at
frequencies o f 0.1, 0.2, 0.4, 1,0, 4.0, 10.0, 100.0, 400.0, 1000.0, and 10,000.0 kHz
using HP LCR meters14. The dielectric constant, K, was calculated from the measured
capacitance, C, using the following equation:
Model Philips 420 STEM, Philips Electric Instruments
Models 4275A and 4274A, Hewlett-Packard, Inc.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where A is the electroded area and t is the sample thickness.
Temperature dependent dielectric measurements were performed to determine the
temperature related transition parameters o f the BaTi03 ceramics (for example, Tc and peak
K). The measurements were carried out using an automated system consisting o f a HP
LCR meter interfaced with a HP desk top computer, in conjunction with a computer
controlled temperature chamber15.
High field polarization and strain hysteresis measurements were carried out using a
computer controlled modified Sawyer Tower system with a National Instruments Input
Output card and a linear variable displacement transducer (LVDT) sensor driven by a lock
in amplifier16, respectively. The voltage was supplied using a voltage DC amplifier17.
Through the LVDT sensor, the strain of the samples can be measured with the application
of an applied field. Electric fields as high as ~ 20 kV/cm were applied using an applied
waveform at 0.2 Hz. During measurements the samples were submerged in Fluorinert18,
an insulating liquid, to prevent arcing.
6.7 Microwave Frequency Measurements
In the frequency range between 50 MHz and 1 GHz, the lumped impedance (lumped
element) technique was employed using an RF impedance analyzer19. The sample holder
consisted o f a coaxial line of Z„ = 50 £2. The center conductor had a diameter of 3.04 mm
while the outer conductor had a diameter o f 7.00 mm. Calibration at the measurement
plane was achieved using three known standards: a short (0 Q), an open (0 S), and a
15
16
17
18
Model MK 2300 Delta Design Inc.
Model SR830, Stanford Research Systems
Model 609C-6, Trek
Model FC -40, 3M
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135
matched load (50 £2). Sample dimensions were generally kept at ~ 3.0 mm diameter and
ranged from 0.4 mm to 1.0 mm in thickness. All samples were gold electroded on their
opposing faces and placed in a drying oven at 150° C for at least 1 hour. Upon removal
from the drying oven, the samples were allowed to equilibrate to room temperature just
before measurement. After calibration, the sample was placed on the end of the center
conductor o f the coaxial line, and backed by a short circuit as illustrated by figure 6.1. The
RF analyzer then measured directly the magnitude, r(0 )|, and phase angle, 9, from
which, dielectric properties were calculated using equations (3.40) and (3.41).
Movable
plunger
Sample
50 £2 Coaxial
Line
J
Inner conductor
Figure 6.1. Illustration of sample holder used for lumped impedance measurements.
Model 4191A RF Impedance Analyzer, Hewlett Packard, Inc.
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136
The cavity perturbational technique was used at several selected frequencies above 1
GHz. Using equation (3.102), the dimensions for cavities resonating in theTM,)10 m ode
possessing resonant frequencies at 1.5, 3.0, and 5.5 GHz, were determined. Three brass
cavities of appropriate dimensions were fabricated with removable top lids, and adjustable
bottom plates. Figure 6.2 illustrates the general characteristics of the cavities. The
dimensions and measured resonant frequency of each cavity are shown in Table 6.2. The
inner wall, top, and bottom plates were polished to mirror-like finishes. The top lid was
designed to be removable for insertion and removal of the samples. Although the top lids
were well machined, removal and subsequent replacement of the top lid did affect the Q o f
the cavity to a small degree. The Q of each cavity was always measured at the beginning o f
a series o f measurements and is presented with the data. Coaxial cables terminated in a
small loop were used to couple into the TM 0I0 mode.
Input/Output Coupling Loop
Removable Lid
Thin M ica Strip
Sample
zzz
Adjustable Bottom Plate
Adjustable Plunger
Figure 6.2. Cross-section o f cavity and sample configuration used for cavity perturbation
measurements.
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137
Table 6.2 Radii and Resonant Frequencies o f T M ^ Cavities
Cavity Radius
(cm)
7.65
3.79
2.03
fo
(GHz)
1.51
3.01
5.60
The general technique requires the measurement of the frequency response of the power
transmission coefficients (S21) for the empty and filled cavity. The frequency response o f
the power transmission coefficients were measured using a Vector Network Analyzer20.
The Network Analyzer is a two-channel microwave receiver deigned to process the
magnitude and phase of the transmitted and reflected waves from a network. The RF
source is set to sweep over a specified bandwidth, while a four-port reflectometer samples
the incident, reflected, and transmitted RF waves. An internal computer calculates and
displays the magnitude and phase of the S parameters or any other desired parameters
(VSWR, R L, group delay, impedance, etc.).
Atypical measurement sequence included first measuring theTM^o resonant frequency
o f the empty cavity,/,. Next, the Q0 at f 0 was determined. The Q is obtained by analyzing
the half—power bandwidth o f the resonant peak (see figure 3.8 and equation (3.116)).
Explicitly, Q0 is determined from,
ft =
t
4
J2
t
.
(6.3)
Jl
where f 2 is the frequency above f Qmeasured along the resonant peak at the - 3 dB (halfpower) point relative to /0, a n d /, is the frequency b elo w /, measured along the resonant
peak at the - 3 dB point relative to /0. A sample in the shape o f a long thin rod of uniform
cross section is then inserted into the cavity as shown in figure 6.2. Samples varied in
length between 1.1 and 1.6 cm o f cross-sectional areas on the order o f 10° cm2 and 102
20
Model H P8510 Vector Network Analyzer, Hewlett Packard, Inc.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
138
cm2 for ceramic and composite samples, respectively. Some samples were allowed to
equal the height of the cavity while others were kept just less than the height of the cavity.
As to which samples were allowed to equal the height o f the cavity depended upon their
dielectric properties. Generally, the higher K ceramic as well as the more lossy samples
(ceramic and composite) required the smallest cross-sectional areas in order to conduct
measurements. When they were kept equal to the cavity height, they often broke on
reclosing the cavity; consequently, samples slightly less than the cavity height were
employed in these cases. Samples less than the height o f the cavity were held
symmetrically along the axis of the cavity in a hole drilled through a thin strip o f mica. The
thin strip of mica had a negligible effect on the resonant properties of the cavity. After
insertion of the sample, and reclosing the cavity, the resonant frequency,/, and Q, of the
filled cavity were measured in the same way as above. Depending on the sample
configuration being employed, the dielectric properties were calculated from either
equations (3.130) and (3.131) or (3.133) and (3.134).
The post resonant technique was used on samples possessing relatively low dielectric
losses (tan8 < 0.02). The technique involved fabricating right cylindrical rods of suitable
dimensions. The posts were then placed between two parallel conducting brass plates.
Signal input and output coupling to the sample was obtained by 0.085 inch semirigid
coaxial cables terminated with small circular loops. The measurement configuration is
shown in figure 6.3. TheTE0ll resonant mode was determined by use of the mode chart
(figure 3.11) and by purposely changing the boundary conditions; that is, the top plate was
slighdy tilted, introducing a small air gap between the sample and conducting plate. This
deviation from the boundary conditions resulted in the adjacent TM modes moving to
higher frequencies, while theTE0I1 mode decreased in Q and moved to a slightly lower
frequency. Once theT E 01l mode was identified,/, and Q0 were determined, from which
the dielectric parameters were obtained from equations (3.140), (3.141) and (3.144).
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139
A filled rectangular X—band waveguide was used to obtain dielectric properties between
8.2 and 12.4 GHz. A section o f waveguide material was filled with the composite material,
from which, a thin flange was cut. The flange was then machined parallel. The network
analyzer was used to measure the frequency response o f the S n and S2I coefficients. To
do this, X-band wave launcher adapters21 were connected to both ports o f the network
1
3
analyzer. A full calibration was conducted at each port using a —X offset short, a - X
8
8
offset short, a fixed matched load and a sliding load. After calibration, the filled waveguide
flange was screwed firmly between the adapters. From the measured response of the S„
and S21 parameters, and equations (3.151), (3.152) and (3.153), the dielectric properties
were determined.
Metal Plate
Input/Output
Coupling Loops
Sample
//-field
Figure 6.3. Illustration o f post resonance (Hakki-Coleman) measurement configuration.
Maury Microwave Corp.
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140
To 8510 Port 1
T o 8510 port 2
Coaxial Waveguide
Adapters
Sample
Port Extension
Cable
I I
Calibration
Planes
Port Extension
Cable
Figure 6.4. Illustration o f rectangular X—band waveguide measurement setup.
■ ib
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CHAPTER VII
RESULTS AND DISCUSSION
In this chapter, the results o f the material characterization will be presented, first for the
ceramics, then for the composites. Following the characterization results for each series of
samples, their high frequency properties will be presented and discussed. An attempt is
made to correlate the measured dielectric spectra with the material characterization results
and observed microstructures. Relaxation parameters, such as the change in dielectric
constant, AAT, and the relaxation frequency, f r, will be used to calculate domain widths and
spontaneous polarization values, through use of the appropriate relaxation models
presented in Chapter IV.
7.1 Ceramic Characterization and Dielectric Spectroscopy
In this section, the results of the ceramic characterization will be presented. Scanning
electron microscopy was conducted in order to observe domain structures and grain sizes in
the various ceramics. Following this, x—ray analysis was carried out for the determination
of lattice parameters and degree of tetragonality. Next, temperature dependent dielectric
measurements were used to determine the transition parameters o f the various ceramics. E—
field dependent hysteretic properties were also examined in each ceramic material. Finally,
the dielectric spectra of the various ceramics were determined through microwave
frequencies.
7.1.1 Ceramic Microstructural Characterization
Ceramic samples of various grain sizes (coarse, small, and fine) were prepared as
outlined in Chapter VI, section 6.2. These samples were processed so as to achieve
microstructures characterized by the following three different grain size regimes: ~ 10.0
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142
pm, ~ 1.0 pm , and ~ 0.1 pm. To distinguish between these various samples the following
identification scheme will be employed: the coarse grain BaTi03ceramics ( ~ 10.0 pm)
will be designated as CGBT; the small grain B aT i03 ceramics (~ 1.0 pm) will be
designated as SGBT; and the fine grain B aT i0 3 ceramics (~ 0.1 pm) will be designated as
FGBT. The densities of the above samples were determined using the Archimedes
technique and are reported in Table 7.1 along with their percent theoretical values.
Table 7.1 Measured Densities of BaTiQ, Ceramic Samples
Sample
Density
Percent Theoretical Density
(g/cc)
CGBT
5.70
94.7
SGBT
5.94
98.7
FGBT
5.89
98.0
Scanning electron microscopy (SEM) was used to examine the ceramic microstructures
in order to characterize the domain structures and grain size distributions of the ceramic
samples. Mean grain size, and standard deviation were determined by analyzing the
photomicrographs using the line intercept method, whereby the grain sizes were fitted to a
gaussian distribution, utilizing commercially available image analysis and statistical
software. Figure 7.1 shows the microstructure o f a polished and chemically etched cro sssection of the CGBT sample. The regions characterized by the series of parallel lines are
due to an alternation between two positions o f the c-axis, differing by 90° in their
orientation (90° domains). The “herringbone” pattern apparent in the upper right hand
comer of figure 7.1, results from the combination of two such alternating domains. The
grain size analysis yielded a mean grain size, p ^ , of 14.45 pm with a standard deviation,
(Tgs, of 8.77 pm.
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143
Figure 7.1. Photomicrograph o f domain structure and grain sizes in coarse grain
polycrystalline BaTi03 (CGBT) ceramic after polishing and etching.
Figure 7.2 illustrates the microstructure representative of the small grain (SGBT)
samples. The groups o f parallel lines in figure 7.2 (a) illustrate the two dimensional
adjustment due to the 90° twinning associated with stress relieve, in accordance with that
predicted for grain sizes less than 10 pm, as discussed in Chapter II, section 2.2.6.2.
Figure 7.2 (b) illustrates the grain size observed in a thermally annealed sample. The
thermal annealing provided better resolution of the individual grains. From this, and
similar photomicrographs, p gs was determined to be 2.14 pm with a CTgs of 1.27 pm.
Figure 7.3 illustrates the microstructure of a thermally annealed FGBT sample;
microstructures of chemically etched samples were somewhat ambiguous, showing no
clear domains or well defined grain boundaries. The grain size analysis yielded a p ^ of
0.26 pm with a a gs of 0.13 pm.
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144
Figure 7.2. SEM photomicrograph of small grain polycrystalline B aT i03 (SGBT) ceramic,
(a) Chemically etched sample showing domain relief, (b) Thermally etched sample
showing grain sizes.
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145
Figure 7.3. SEM photomicrograph o f fine grain polycrystalline B aT i03 (FGBT) ceramic,
thermally etched sample showing grain sizes.
Table 7.2 summarizes the calculated mean grain size and standard deviation obtained from
the observed microstructure of each ceramic.
Table 7.2 Grain Size Statistics of BaTiQ, Ceramic Samples
Sample
(|im)
(dm)
CGBT
14.45
8.77
SGBT
2.14
1.27
FGBT
0.26
0.13
7.1.2 Ceramic X—ray Diffraction Analysis
X-ray diffraction (XRD) analysis was used to determine lattice parameters and degree
of tetragonality o f each ceramic specimen. Lattice parameters were determined by the
Cohen’s least-square analytical method using a computer program. Diffraction peaks were
obtained in the range of 20 = 20° to 80° and peak position was determined by analyzing the
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146
individual peaks using a Lorentzian curve fit function. Figure 7.4 shows the {200}
reflections obtained from the various ceramic samples. The CGBT sample shows obvious
tetragonality, as evident from the distinct splitting of the {200} reflections. In figure 7.4,
the intensity of the (002) reflection is greater than the sum of the intensities for the (200)
and (020) reflections. This is attributed to preferred domain orientation introduced at the
surface of the ceramic due to a two-dimensional tensile stress introduced as a consequence
of polishing and machining the samples (Jyomura, 1980). The SGBT sample also shows
a strong indication of tetragonality as well. The structure of the FGBT ceramic, is at best,
inconclusive. Figure 7.4 supports that the structure is tending towards cubic as the degree
of splitting of the {200} reflections is greatly reduced, however, it is expected that for a
purely cubic structure, the
peak would be more clearly resolved. Consequendy, the
FGBT sample can be best described as pseudocubic. The decrease in tetragonality is
attributed to the effects of internal stress on the crystal structure. In the coarse grain
ceramic, CGBT, the large grain size allows for the development of the banded domain
structure which provides a three dimensional compensation for homogenous stress. The
small grain ceramic, SGBT, allows for some internal stress relief within the grains, with
the development of inhomogeneous stress at the grain boundaries. It is plausible that in the
FGBT material, the grain size is too small for domain twinning to occur, resulting in high
internal stresses through out the grain. This uncompensated stress is expected to reduce the
tetragonal distortion, leading to a pseudocubic phase. A summary of the calculated lattice
parameters is provided in Table 7.3.
Ceramic
CGBT
SGBT
FGBT
•it-1
Talale 7.3 Summary of Ca culated Lattice Parameters
a lattice parameter
c lattice parameter
(A)
(A)
3.992
4.033
3.991
4.027
4.015
4.015
_______________
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d a ratio
(A)
1.010
1.009
1
147
FGBT
'002
SGBT
200
44.0
44.5
45.0
45.5
CGBT
46.0
46.5
20
Figure 7.4. {200} reflections from XRD patterns obtained on the various ceramics.
*i£^
_ _______ ______
R e p ro d u c e d with p e rm iss io n of th e co p y rig h t ow n er. F u rth e r re p ro d u c tio n pro h ib ited w ith o u t p e rm issio n .
148
7.1.3 Ceramic Temperature Dependent Dielectric Measurements
Low frequency (100 Hz to 10 MHz) room temperature and temperature dependent (-100
to 180 °C) dielectric measurements were conducted on each ceramic specimen. Room
temperature measurements revealed a sharp increase in the dielectric constant for the
ceramics o f average grain size equal to ~ 2 pm, consistent with that reported in literature. A
summary of the room temperature dielectric data obtained at 10 kHz is presented in Table
7.4. Figure 7.5 better illustrates the trend in K and tan5 with varying grain size. In the
grain size range between 14 pm and 2 pm, the observed strong increase in K with
decreasing average grain size is ascribed to an increased density of domain walls which
results in an increasing contribution to the permittivity. At some grain size below ~ 2 pm ,
the overall internal stress energy is no longer reduced by twinning, consequently, the
number of domain walls decreases and so must their contribution to the permittivity.
Similar arguments explain the behavior in tan5.
Table 7.4 Ceramic Room Temperature Dielectric Data Obtained at 10 kHz
Sample
CGBT
SGBT
FGBT
Hgs
14.45
2.14
0.26
Dielectric Constant
1998
6290
2071
tan5
0.00575
0.026105
0.00634
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149
7000
tanS
5600
0.08
4200
0.06
2800
0.04
1400
0.02
ees
c/a
c
o
U
u
•c
gtjj
u
Q
0
10
0.1
100
Average Grain Size (pm)
Figure 7.5. Average grain size effect on the room temperature dielectric properties of
ceramic B a T i0 3 at 10 kHz.
The temperature dependent behavior of the dielectric constant at 10 kHz, 100 kHz, and 1
MHz for each o f the samples CGBT, SGBT, and FGBT is shown in figures 7.6, 7.7, and
7.8, respectively.
.2 10
lO.kHz
100 kHz
1 MHz
8000
6000
4000
2000
0
-100
-50
0
50
100
150
Temperature (°C)
Figure 7.6. Temperature dependence in the dielectric constant of CGBT.
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200
150
1.2 10"
10 kHz
8000
100 kHz
u
5
6000
a
4000
2000
-100
0
-50
50
100
150
200
Temperature (°C)
Figure 7.7. Temperature dependence in the dielectric constant of SGBT.
3000
10 kHz
100 kHz
1 MHz
2500
2cy.
C
CJ
2000
1500
1000
-100
-50
0
50
100
150
Temperature (”C)
Figure 7.8. Temperature dependence in the dielectric constant of FGBT.
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200
151
Figures 7.6 through 7.8 consistently illustrate frequency dependent dispersion in the
dielectric constant below Tc. At temperatures above Tc, the frequency dependent dispersion
is absent up to 1 MHz; above 1 MHz, again frequency dependent dispersion was apparent
in all ceramic samples. Also, figures 7.6 through 7.8 clearly exhibit the high-temperature
ferroelectric to paraelectric transition, characterized by the sharp peak in the dielectric
constant. The CGBT and SGBT samples also showed peaks associated with the lower
temperature phase transitions. The peaks in the dielectric constant associated with the
lower temperature transitions were somewhat ambiguous in the FGBT sample. It appears
that a broad peak centered near - 30 °C, indicative of a diffuse phase transition, is present.
A summary of the transition parameters for the ceramic BaTi03 samples is provided in
Table 7.5.
Sample
CGBT
SGBT
FGBT
Temperature
(C ubicTetragonal)
re)
127
124
119
Curie Constant
CC)
132896
161361
156937
Temperature
(TetragonalOrthorhombic)
(°C)
15
20
-3 0
Temperature
(OrthorhombicRhomboheral)
CC)
-90
-76
—
Figure 7.9 compares the temperature dependence of the dielectric constant observed in
each sample at 1 MHz. In figure 7.9, it can be seen that above the Tc, the dielectric
properties of both the CGBT and SGBT samples coincide, whereas the dielectric properties
of the FGBT differs dramatically from the larger grain samples. The difference in the
behavior of the dielectric constant observed in the FGBT above Tc relative to the other
samples, as well as the apparent broadening of the lower temperature transitions, is again
attributed to stress related crystallographic changes.
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152
C
u
u
•y i
SGBT
8000
6000
ZJ
a
FGBT
4000
2000
-100
-50
0
50
100
150
200
Temperature CC)
Figure 7.9. Comparison between the temperature dependence of the dielectric constant of
ceramic BaTiO. samples o f different average grain size.
7.1.4 Ferroelectric Hysteretic Properties
Polarization versus electric field and strain versus electric field were measured from the
various ceramics, and are shown in figures 7.10 and 7.11. The remanent polarization, Pr,
was ~ 6 pC/crrr for the CGBT, increased to ~ 12 p.C/cm2 for the SGBT, and then dropped
to ~ 2 (IC/cm2 for FGBT. The Pr appears to follow the expected domain wall density in
these materials, with the SGBT having the highest Pr. The low Pr o f the FGBT may be
interpreted as a consequence o f the low density of domain walls, as well as the clamping of
any existing domains due to the large unrelieved internal stresses.
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153
(a)
-20
-15
-10
-5
0
5
Electric Reid (kV/cm)
10
15
20
(b)
30
20
e
SGBT
PGBT
c
r3
N
)
L.
a
cu
-10
-20
-30
-80
-60
-40
-20
0
20
40
60
80
Electric R eid (kV/cm)
Figure 7.10. Dielectric hysteresis measured in (a) CGBT and (b) SGBT and FGBT.
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154
0.5
0.4
FGBT
SGBT
0.3
CGBT
0.2
0.1
0
0. 1
-80
-60
-40
-20
0
20
40
60
80
Electric Field (kV/cm)
Figure 7.11. Strain versus electric field for CGBT, SGBT, and FGBT.
7.1.5 Dielectric Spectroscopy of Ceramic BaTiO, Samples
The dielectric properties (dielectric constant and dissipation factor) of the various
BaTiO, samples were measured through microwave frequencies at room temperature using
the techniques described in Chapter VI. In the case o f the CGBT, a number of bulk
samples were available, from which numerous samples could be prepared. Consequently,
at least five or more samples were measured at each frequency. From these measurements
conducted on multiple samples at each frequency, an average was taken. This is the
reported value in all subsequent Tables and figures. The error presented along with the
average measured values was determined from the maximum and minimum measured
values from any set of samples at a particular frequency. In the case of the SGBT and the
FGBT, the availability of bulk samples was limited. Although the hot pressing process
used to fabricate these samples resulted in high densities, the sample experienced multiple
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155
fractures. These fragments were carefully cut and machined to yield usable samples, from
which all subsequent characterization and dielectric measurements were obtained.
However, due to the limitation in the availability o f large bulk samples, only a limited
number of samples could be prepared for microwave measurements. Another concern was
the fragility o f the ideally prepared samples for high frequency dielectric measurements.
Particularly in the case of the cavity perturbation method, the ideal geometry demands long,
very thin samples. However, in the case o f the hot pressed samples, this condition was
relaxed, so as to ensure that measurements could be conducted before the sample broke.
The above problems and concerns resulted in the collection of dielectric data from just one
or two samples each from the SGBT and FGBT samples.
Low frequency (100 Hz to 10 MHz) measurements were conducted using the method
discussed in chapter VI, section 6.5. Table A shows the room temperature data measured
from the CGBT samples in the frequency range of 100 Hz to 10 MHz.
Table 7.6 Room Temperature Low Frequency Dielectric Properties of CGBT
F
Frequency
+K
-K
^average
+tan5
-tanS
t^n^average
100 Hz
2099.5
85.29
80.70
0.0416
0.00255
0.00325
200 Hz
2065.5
85.81
74.31
0.0295
0.00275
0.00272
400 Hz
2047.3
84.54
70.93
0.0195
0.00168
0.00231
1 kHz
2030.6
83.43
68.23
0.0135
0.00150
0.00175
4 kHz
2013.3
82.07
64.83
0.00823
0.00106
0.00153
10 kHz
1997.9
94.04
74.48
0.00605
0.00132
0.00108
100 kHz
1982.2
94.07
79.32
0.00311
0.00103
0.00076
400 kHz
1977.4
95.12
83.91
0.00475
0.00165
0.00125
1 MHz
1973.4
94.34
85.59
0.00831
0.00410
0.00204
10 MHz
1952.0
90.38
87.00
0.05800
0.04104
0.03363
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156
In Table 7.6, ATaverage is the average measured value o f AT, +K is the difference obtained
by subtracting ATaverage from the maximum measured AT, -ATis the difference obtained by
subtracting the minimum measured K from ATaverage, where +tanS and -tan8 are obtained
in similar ways.
Similarly, a lumped impedance technique was used to obtain dielectric data in the
frequency range between 50 MHz and 1 GHz using the technique outlined in Chapter VI,
section 6.6. Table 7.7 illustrates typical raw data for a representative CGBT sample having
a diameter of 2.96 mm and a thickness of 0.272 mm.
Table 7.7 Representative Raw Data and Calculated K and tanS From Lumped Impedance
Measurements Conducted on CGBT
Frequency
K
tanS
e
f(0 )|
(MHz)
(degrees)
0.9807
50
-162.36
1823.80
0.0643
100
0.9802
-171.11
1797.63
0.1294
200
0.9807
-175.54
1716.48
0.2507
300
0.9825
-177.06
1649.95
0.3443
400
0.9837
-177.85
1588.05
0.4382
500
0.9849
-178.34
1537.20
0.5254
600
0.9858
-178.78
1532.44
0.6719
700
0.9842
-179.13
1272.70
1.0430
800
0.9864
-179.40
1251.40
1.3080
900
0.9863
-179.65
847.03
2.2600
1000
0.9860
-179.90
245.30
8.0940
Table 7.7 also illustrates one o f the fundamental problems associated with this technique in
the measurement o f high-Ar materials. Recall that T(0) = -1 for the case where the load is
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157
a short circuit. Written in terms o f its magnitude and phase angle,
f(0 ) = -1 => |f(0 )| = 1, 0 = —180.0°. Note that at the highest frequencies, this condition
is rapidly approached, particularly in the value of 0. Practically, this means that the
measured values become more difficult to distinguish from that of the short circuit used in
the calibration procedure. The consequence of this is the incorporation of increasingly
larger errors at higher frequencies. These errors are then manifested in the calculation of
the dielectric constant and dissipation factors. Because o f this measurement limitation, the
lumped impedance technique was used up to a maximum frequency of 600 MHz for the
ceramic samples. Table 7.8 summarizes the results obtained from lumped impedance
measurements on five different CGBT samples of dimensions characterized by diameters
between 3.01 and 2.92 mm, and thicknesses between 0.612 and 0.184 mm.
Table 7.8 Room Temperature
Frequency
+K
“ “average
(MHz)
50
1785.51
91.75
-K
+tan5
-tanS
81.55
t^^average
0.0628
0.00507
0.01293
100
1757.95
74.780
86.040
0.1262
0.00873
0.02180
200
1681.34
87.010
98.520
0.2434
0.01907
0.04021
300
1609.96
114.44
119.85
0.3351
0.04050
0.07795
400
1508.30
79.750
101.82
0.4578
0.03959
0.01959
500
1439.08
98.120
104.83
0.5603
0.06623
0.03489
600
1388.93
143.51
105.35
0.7373
0.07617
0.10703
A cavity perturbation technique was used for the measurement of dielectric properties at
selected frequencies above 1 GHz. A TM^,, resonating cavity was employed as outlined in
Chapter VI, section 6.6. To carry out these measurements, samples were fabricated in long
thin bars o f approximately square cross sections. The volume o f each bar was determined
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
by carefully measuring its length, t , the thickness o f one side at two positions along its
length (5 , and 52), and then taking an average, and the thickness of the other side at two
positions along its length O ' and s'2), and taking the average. Tables 7.9 through 7.11
summarize the measured parameters and calculated dielectric properties. In Tables 7.9
through 7.11,/0 is the resonant frequency o f the empty cavity; Q0 is the quality factor o f the
empty cavity; Vs is the volume of the sample; V0 is the volume o f the cavity;/, is the
resonant frequency o f the filled cavity; Qx is the quality factor o f the filled cavity; IL is the
insertion loss of the filled cavity; and
is the relative shift in the resonant frequency
upon filling the cavity.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R eproduced with permission
of the copyright ow ner.
Table 7.9 Measured Parameters and Calculated Dielectric Properties of CGBT Using Cavity Perturbation at 1.5 GHz
Further reproduction
/ 0= 1.5116 GHz, Q0 = 1046-1089
Sample
C
1,2 average
(cm)
c9
I, 2 average
I
Vo
/,
(cm3)
(GHz)
Q^
IL
4 f //o
K
tan8
(cm)
x 10°
(cm3)
1.592
3.4407
290.3
1.4831
53
-62.1
0.019
949.5
0.4748
m
prohibited without p e rm issio n .
1
0.04750
(cm)
0.0455
2
0.0460
0.0455
1.593
3.3341
290.3
1.4866
62
-60.8
0.017
855.7
0.4576
3
0.0435
0.04650
1.593
3.2220
290.3
1.4864
58
-61.2
0.017
893.3
0.4951
4
0.0450
0.0475
1.592
3.4029
290.3
1.4862
59
-61.2
0.017
849.7
0.4814
5
0.0420
0.0465
1.592
3.1092
290.3
1.4866
60
-61.2
0.017
915.5
0.4801
in
vO
R eproduced with permission
of the copyright ow ner.
Table 7.10 Measured Parameters and Calculated Dielectric Properties of CGBT Using Cavity Perturbation at 3 GHz
s'1, 2average
i
(cm)
Q^
2.9422
113
V0
/.
(cm)
v<
x 103
(cm3)
(cm3)
(GHz)
1.593
3.3704
73.4
A ///0
K
tan8
-58.2
0.027
333.6
0.1587
prohibited without p e rm issio n .
1
0.0455
(cm)
0.04^5
2
0.0455
0.0430
1.590
3.1108
73.4
2.9348
90
-59.5
0.029
399.6
0.1891
3
0.0465
0.0440
1.591
3.02552
73.4
2.9488
134
-56.7
0.024
313.0
0.1434
4
0.0465
0.0460
1.592
3.4053
73.4
2.9297
98
-59.9
0.031
380.4
0.1623
5
0.0420
0.0465
1.592
3.1092
73.4
2.9454
126
-57.3
0.025
346.1
0.1468
091
c
1,2average
•*.
r-
Sample
^
cd
Further reproduction
/ 0 = 3.0227 GHz, Q0 = 1284
R eproduced with permission of the copyright ow ner. Further reproduction
Table 7.11 Measured Parameters and Calculated Dielectric Properties of CGBT Using Cavity Perturbation at 5.6 GHz
f 0 = 5.5942 GHz, {?„ = 275
Sample
c
1,2average
(cm)
sf
1, 2average
£
V0
/,
(cm3)
(GHz)
IL
A ///0
K
tanS
(cm)
V ,,
x 10 3
(cm3)
1.248
2.6114
16.7
5.1975
45
-26.9
0.071
279.3
0.1482
Q,
(dB)
prohibited without p e rm issio n
1
0.0465
(cm)
0.04^0
2
0.0445
0.0460
1.247
2.5526
16.7
5.2185
54
-25.7
0.067
269.2
0.1248
3
0.0450
0.04650
1.247
2.6093
16.7
5.2103
51
-26.1
0.069
269.9
0.1315
4
0.0450
0.0460
1.250
2.5875
16.7
5.2020
49
-26.4
0.070
276.0
0.1340
5
0.0460
0.0450
1.249
2.5854
16.7
5.1737
40
-27.8
0.075
300.7
0.1610
o\
162
From the data shown in Tables 7.9 through 7.11, the average dielectric properties along
with their associated error are shown in Table 7.12.
______ Table 7.12 Room Temperature Cavity Perturbation Measurements on CGBT
Frequency
+K
-K
average
-tanS
+tan5
tan8avcragc
(GHz)
1.48
892.7
43.050
0.4778
0.01730
0.02020
56.800
2.94
356.3
43.310
40.050
0.1600
0.02910
0.01114
5.20
279.04
21.640
9.7900
0.1399
0.02111
0.01509
Using the data shown in Tables 7.6, 7.8 and 7.12, the spectrum of dielectric properties for
the CGBT sample was plotted as a function of the logarithmic of frequency. These results
are shown in figure 7.12.
In figure 7.12 the dielectric constant of CGBT decreases from its low frequency value
of ~ 2000 measured at 10 kHz to its high frequency value of ~ 300 measured at ~ 5 GHz.
The dielectric loss gradually increases from its low frequency value of 0.006 measured at
10 kHz, traverses a maximum o f ~ 0.8 at 771 MHz, and rapidly decreases to ~ 0.1 at ~ 5
GHz. The A K is thus ~ 1700, and by defining the frequency at which tan6 peaks as the
relaxation frequency,/., for CGBT / = 771 MHz. These relaxation parameters (AAT and
/ ) may now be quantitatively related to Ps and domain width, d, using the theories outlined
in Chapter HI. Because the relaxation spectrum appears to show no resonant character, and
in addition, since microstructural analysis as well as predictive models suggest a
predominance of 90° domain twinning, the relaxation model connecting shear wave
emission by 90° domain walls is emphasized here. This relaxation model detailed in
Chapter IV, section 3.1 allows for the calculation of Ps from AK using equation (4.29) and
the calculation of the domain width, d, from/ , using equation (4.31). Using the A K of
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1700, c55 = 34.02 GN/m 2 and S0 = 0.01, Ps was calculated to be 32 pC/cm 2. This value is
high compared to the expected value o f 26 pC/cm2. A possible explanation for this
discrepancy may be that the CGBT samples have an average grain size o f 14.4 pm. This
size is near the grain size (~ 10 pm) where it is expected that the twinning associated with
the three dimensional stress compensation is hindered. At this average grain size, it is
likely that the 90° twinning allowing only for two dimensional stress relief is also
occurring. As previously discussed, these twinning patterns are highly dependent on grain
size, leading to anomalously high room temperature ATs, generally in the grain size range
below ~ 10 pm, with K peaking at a grain size o f 1-0.7 pm (see figure 4.17). In other
words, at grain sizes o f - 10 pm and below, the domain twinning pattern and domain
width are entering into a strong dependence on grain size. Its possible that there may be
some marginal increase in the low frequency K due to this effect, which would result in a
higher AK. This seems to be the reason for the anomalously high calculated Ps.
From the measured relaxation frequency and equation (4.31), a domain width of 0.98
pm was calculated. This width can be compared with that observed in the microstructure
of the CGBT sample as illustrated in figure 7.1. It does appear that many of the domains in
figure 7.1 are on the order o f 1 pm.
with permission of the copyright owner. Further reproduction prohibited without permission.
R eproduced with permission of the copyright ow ner.
2500
2000
1500
U
o
■ f i
jj
1000
prohibited without p e rm issio n .
Q
0.5
500
Dielectric Loss, tan5
Further reproduction
§
o
--Q -Q -a ^ 6 ^ o ^ >
2
3
4
5
6
7
8
9
10
log/ (Hz)
Figure 7.12. Spectrum of dielectric properties measured from CGBT ceramics.
£
165
In a similar way, the dielectric properties of the SGBT samples were measured. First,
low frequency room temperature measurements were conducted and are presented in Table
7.13. Unfortunately, only a few samples survived all of the cutting and machining
operations necessary for subsequent measurements. For this reason, only two samples
were available for low frequency measurements with only one available for lumped
impedance measurements.
Table 7.13 Room Temperature Low Frequency Dielectric Properties of SGBT
Frequency
+K
-K
^average
+tan8
-tanS
tan8average
100 Hz
6620.0
332.0
332.0
0.00050
0.00050
0.0247
200 Hz
6565.1
329.5
329.5
0.0236
0.00070
0.00070
400 Hz
6516.9
325.7
325.7
0.0223
0.00078
0.00078
1 kHz
6455.1
317.9
317.9
0.0227
0.00079
0.00079
4 kHz
6361.3
310.4
310.4
0.0239
0.00094
0.00094
10 kHz
6289.9
305.1
305.1
0.0261
0.00109
0.00109
100 kHz
6089.2
284.9
284.9
0.0375
0.00170
0.00170
400 kHz
5871.2
295.9
295.9
0.0544
0.00450
0.00450
1 MHz
5674.2
273.3
273.3
0.0735
0.00725
0.00725
10 MHz
4925.49
201.6
201.6
0.1665
0.03050
0.03050
The results of the lumped impedance measurements are shown in Table 7.14. Because of
the high K and lossy nature of these samples, lumped impedance measurements could only
be relied upon up to 300 MHz. Above this frequency, large errors became apparent due to
the load mimicking a short circuit.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
166
Table 7.14 Room Temperature Lumped Impedance Measurements on SGBT
Frequency
K
tanS
(MHz)
50
3789.4
0.2328
100
3463.9
0.2893
200
3333.7
0.3841
300
3550.6
0.5371
Cavity perturbation measurements were attempted on a limited number o f these samples,
however, in every case, the shifted peak of the filled cavity was just too attenuated to be
resolved through the background noise. For this reason, no dielectric data above 300 MHz
could be collected on these samples using the methods outlined in this work. The low
frequency and limited lumped impedance measurement results as well as the problems
encountered in the measurement of the dielectric properties above 300 MHz suggest that the
high—AT lossy behavior of these samples persist well into the microwave frequency range.
Using the data shown in Tables 7.13 and 7.14, the spectrum of dielectric properties for the
SGBT sample was plotted as a function of the logarithmic of frequency. These results are
shown in figure 7.13. In figure 7.13, evidence of dielectric relaxation is clearly present,
whereby the dielectric constant decreases from its low frequency value with an associated
increase in the dissipation factor. The dielectric constant does appear to be approaching a
saturation value at the highest measured frequency, however, that is likely due to associated
measurement errors, and is not believed to be a true saturation of the dielectric properties.
If the dielectric constant were truly saturating to its high frequency value at 200 MHz, it is
expected that the dissipation factor would have exhibited a loss peak a some lower
frequency. This, as illustrated in figure 7.13, is not the case.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6500
*
Further reproduction
C/3
c
O
U
o
6000
Dielectric Loss, tan5
R eproduced with permission of the copyright ow ner.
7000
5500
5000
• f i
Si
D
4500
prohibited without p e rm issio n
4000
0.5
tanS
3500
3000
2
3
4
5
6
7
8
9
10
lo g / (Hz)
Figure 7.13. Spectrum of dielectric properties measured from SGBT ceramic.
ON
168
Finally, the dielectric properties of the FGBT samples were measured. First, low
frequency room temperature measurements were conducted and are presented in Table
7.15. Unfortunately, only a few samples survived all of the cutting and machining
operations necessary for subsequent measurements. For this reason, only one sample was
available for low frequency measurements with two samples available for lumped
impedance measurements.
Table 7.15 Room Temperature Low Frequency Dielectric Properties of FGBT
F
Frequency
^average
t^n^averagc
100 Hz
2117.9
0.0071
200 Hz
2109.6
0.0065
400 Hz
2103.1
0.0053
1 kHz
2093.2
0.0060
4 kHz
2081.3
0.0054
10 kHz
2071.4
0.0063
100 kHz
2049.4
0.0082
400 kHz
2034.1
0.0112
1 MHz
2020.7
0.0136
10 MHz
1966.7
0.0319
Because of the measurement limitation problems previously discussed, the lumped
impedance technique was used up to a maximum frequency of 700 MHz for the ceramic
FGBT samples. Table 7.16 summarizes the results obtained from the lumped impedance
measurements on two different FGBT samples.
•ci _ ________
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
169
Table 7.16 Room Temperature Lumped Impedance Measurements on FGBT
IT
Frequency
+K
-K
average
-tan5
+tan5
t^nSaverage
(MHz)
50
0.00794
1963.46
41.5
41.5
0.0473
0.00794
100
1938.53
12.7
12.7
0.0727
0.01526
0.01526
200
1924.57
12.8
12.8
0.1141
0.02780
0.02780
300
1938.44
16.1
16.1
0.1498
0.03947
0.03947
400
1967.96
19.3
19.3
0.1805
0.05005
0.05005
500
2010.65
23.14
23.14
0.2080
0.05975
0.05975
600
2111.05
49.33
49.33
0.2157
0.05912
0.05912
700
2352.73
244.2
244.2
0.2569
0.08838
0.08838
A cavity perturbation technique was used for the measurement o f dielectric properties at
selected frequencies above 1 GHz. Because of the limited amount of sample, only two thin
rods suitable for measurement could be fabricated. In addition, as discussed above, these
samples were machined to dimensions slightly larger than what was desired so as to ensure
that these samples would not break before subsequent measurements. The results o f the
cavity perturbation measurements are shown in Table 7.17. Again, because multiple
samples were not available, a determination o f the error associated with these perturbation
measurements was not possible.
Table 7.17 Room Temperature Cavity Perturbation Measurements on FGBT
Sample
1
2
v ..
x 10'3
(cm3)
2.8084
3.6815
/,
(GHz)
3.273
4.1211
IL
(dB)
-79.7
-38.6
Ql
6
7
Af//0
0.41
0.26
K
tan5
1136.6
658.9
0.2079
0.2650
Using the data shown in Tables 7.15, 7.16 and 7.17, the spectrum o f dielectric properties
for the FGBT sample was plotted as a function o f the logarithmic of frequency. These
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
170
results are shown in figure 7.14. As seen in figure 7.14, the dielectric properties of the
FGBT samples change very little up to ~ 100 MHz. Up to this frequency, the dielectric
constant decreases only slightly from its low frequency value o f ~ 2100 to ~ 1900 at 100
MHz. Also, the dielectric loss remains relatively flat and low up until 100 MHz; above 100
MHz, the loss increases more dramatically. Above 100 MHz, the dielectric properties
begin to increase rather sharply; the dielectric constant increases, and then, as shown by the
cavity perturbation measurements, undergoes a sharp decrease above 2 GHz. Through this
frequency range, the loss appears to be passing through a maximum. This is indicative o f
definitive resonant behavior through these frequencies. Thus, the spectrum seems closer to
a true resonance than a relaxation. It is expected that at higher frequencies, the dielectric
constant saturates out to its clamped value. The resonant behavior is attributed to the
coupled resonance of piezoelectric single domain grains o f different size and orientation,
relative to the measuring field. If this is the case, the distinctive features in the frequency
dependence o f the dielectric constant and dielectric loss, should be controlled, in large part,
by the majority of grains having a grain size close to the mean grain size (0.26 pm); that is,
the controlling dimension of the fundamental resonating unit is expected to be ~ 0.26 pm .
Because o f the resonant character exhibited in the dielectric properties of the FGBT
samples, it becomes tempting to use equations (4.8) through (4.11), which were developed
to describe the equivalent circuit of a grain around its resonant frequency. Equation (4.12)
becomes,
2 n jL &
Id^JpS
(7.1)
In (7.1),/o is the resonant frequency of the grain of grain size d, and SE is the mechanical
compliance at constant field. The value of SE depends on the orientation o f the grain with
respect to the field and the coupling mode, however, from given values of SE (Landolt—
Bonrstein, 1981),/0 could be placed between 5.1 and 8.5 GHz. These values seem
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
171
slightly high when compared to the observed spectrum, and may be explained by the fact
that the effective length of the resonant units is increased through intergrain coupling,
thereby lowering the effective resonant frequency.
Figure 7.15 compares the measured dielectric spectra of all three ceramic samples.
From figure 7.15 it is obvious that the SGBT sample exhibits the onset of the relaxation
phenomenon at the lowest frequency, - 1 0 0 kHz. At this frequency the dielectric constant
begins to significantly decrease and the loss increases with increasing frequency. Above 100 kHz, the SGBT exhibits the highest loss o f all ceramic samples up to its highest
frequency of measurement. Again, the apparent saturation in K is attributed to
measurement error.
Up to a frequency of - 10 MHz, the dielectric properties of the CGBT and FGBT
samples are virtually equivalent. Above 10 MHz, the CGBT exhibits Debye—like
relaxation, characterized by a sharp peak in the loss at 771 MHz. The dielectric properties
of the FGBT samples differ markedly above 10 MHz. The dielectric constant o f the FGBT
remains essentially flat, showing a slight decrease up to - 400 MHz; above this frequency,
the dielectric constant increases, and as supported by the highest frequency measurements,
traverses a maximum, and then decreases. The FGBT samples also exhibited the lowest
loss of the various ceramic samples, with its loss peak apparently shifted to a higher
frequency, compared to that of the CGBT samples. The loss peak of the FGBT appears to
be centered at - 1.6 GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
172
Dielectric Loss, tan§
«o
©
cn
oo
00
o
cO
cn
CN
©
CN
X *jirejsuo3 otuoopiQ
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
o
ou
Figure 7.14. Spectrum of dielectric properties measured from FGBT
N
R eproduced with permission
S*
*s
C/3
6000
0.8
Further reproduction
* = CGBT, £
*— SGBT,
fc S b t, k
5000
C
o
u
o
4000
• f i
<L)
Q
o
-o
3000
a
- CGBT, tan 5
- SGBT, tan 5
- FCj BT, tan5
prohibited without p e rm issio n
ii
2000
0.6
0.4
>
1
Dielectric Loss, tan6
of the copyright ow ner.
1
0.2
1000
0
1
2
3
4
5
6
7
8
9
10
log/ (Hz)
Figure 7.15. Comparison of dielectric spectra measured from the various ceramics.
u>
174
7.2 BaTiOj Powder and Composite Characterization and Dielectric Spectroscopy
In this section, the results of the B aT i03 powder and BaTiOj powder—polymer
(polypropylene) matrix characterization will be presented. Particle size distribution and
specific surface area measurements were carried out on the powder samples used for
composite fabrication. Microstructural analysis using transmission electron microscopy
(TEM) was also conducted on the hydrothermal powders. X—ray analysis was carried out
for the determination of lattice parameters and degree of tetragonality of the various
powders. Scanning electron microscopy was used in order to observe the particle
distribution and connectivities in the composites. An attempt is made to correlate the
measured dielectric spectra with the material characterization results and observed
microstructures. Relaxation parameters, such as the change in dielectric constant, AK, and
the relaxation frequency, f r, will be used to calculate domain widths and spontaneous
polarization values, through use of the relaxation models presented in Chapter IV.
7.2.1 Particle Size Distributions and Specific Surface Area Measurements
Using the techniques outlined in Chapter VI, section 6.1.1, the particle size distribution
and specific surface area of the various powders were measured. Figures 7.16 (a) and (b)
present the measured particle size distributions (PSD) of the TAM TICON HPB powder,
from here on, designated as the TAM, and the Cabot BT-8 powder, from here on,
designated as just BT-8. The PSD of the Cabot BT-16 powder, from here on designated
as just BT—16, could not be measured using similar techniques due to excessive particle
agglomeration. The PSD of the BT—8 powder (7.16 (b)) revealed the presence of particles
as large as 1 pm. It is believed that the appearance of these larger particles is mainly due to
the presence of agglomerates affecting the distribution. The specific surface area was
measured on all powders. From the surface area measurements, the primary particle size
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
175
(a)
0.05
-
0 .6
-
- 0.04
a
-
0.03
0.4 -
-
0.02
0.2
-
0 .0 1
E
E
U
-
0 .0 1
0.1
1
10
Normalized Probability Density
0.8
0549
100
Particle Size dim)
(b )
100
40
80
40
0.01
10
0.1
Panicle Size (jxm)
Figure 7.16. Particle size distributions o f (a) TAM , and (b) BT-8.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Percent Pass
24
176
was calculated using equation (6.1). A summary of the particle size statistics and surface
area results are presented in Table 7.18.
T ^ le7^ 8_S ^ m m ^ ^ ofP S D _^ ^ S ^ ^ ific_S ^ ^ ^ e_^ ^ aM e^ ^ rc^ n ^
Powder
Mean Particle
Specific Surface Primary Particle
Standard
Size
Area
Size
Deviation
(Um)
(m2/g)
(pm)
0.72 pm
1.37
TAM
1.33
0.77
0.12 pm
8.26
BT-8
0.19
0.17
66.9 nm
—
14.81
BT-16
—
7.2.2 Powder X-ray Diffraction
X-ray diffraction (XRD) analysis was used to determine lattice parameters and degree
of tetragonality of each powder specimen. High angle reflections were carefully examined
for peak splitting, indicative of the tetragonally distorted perovskite structure. Figure 7.17
shows the {220} reflections obtained from the various powders. The TAM powder shows
obvious tetragonality, as evident from the distinct splitting of the {220} reflection. The
BT-8 shows a strong indication o f tetragonality as well, albeit, to a lesser degree than the
TAM powder. Finally, the structure o f the BT-16 powder, is at best, inconclusive. Figure
7.17 supports that the structure is tending towards cubic, however, it is expected that for a
purely cubic structure, the
peak would be more clearly resolved. Figure 7.17 does
support that with decreasing particle size there is a decrease in tetragonality as well. The
BT—16 powder can best be described as pseudocubic, with perhaps a fraction of the
distribution of particles possessing ferroelectricity. A summary o f the calculated lattice
parameters is provided in Table 7.19.
Powder
TAM
BT-8
BT-16
Table 7.19 Summary of Calculated Lattice Parameters
a lattice parameter
c lattice parameter
(A)
(A)
3.988
4.029
3.998
4.012
4.005
4.005
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
d a ratio
(A)
1.010
1.004
1
220
BT-16
BT-8
./
022
220
TAM
65.0
65.5
66.0
66.5
67.0
2 0
Figure 7.17. {220} reflections from XRD patterns obtained on the various powders.
f
_
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
178
7.2.3 Powder Microstructural Characterization Using TEM
An attempt was made to observe any existing domain structure as well as to determine
the morphology of the hydrothermal powders using transmission electron microscopy
(TEM) examination. These samples were small enough to be imaged directly, unlike the
TAM powder, which would have required more sophisticated preparation techniques, prior
to study. Figure 7.18 (a-d) shows the TEM micrographs obtained on the powders using
both bright field and dark field imaging techniques. The morphology of the BT-8 particles
was characterized by irregular aspherical shapes. Relatively large open pores could be
observed along the perimeter of some of the particles. On some o f the smaller particles, it
was possible to observe non-concentric thickness fringes, indicative of the presence of
some small amount of lattice distortion. The exact nature of the lattice distortion could not
be determined, however, possible causes could include dislocations, point defects or the
presence of strain fields. The particles ranged in sizes between ~ 100—200 nm. The B T 16 appeared to be highly agglomerated. Agglomerates ranged in sizes between ~ 100-200
nm. Within the agglomerates, the individual particles could be discerned. These particles
had highly irregular, non-spherical shapes. They ranged in sizes between ~ 30—50 nm. In
this case as well, non-concentric thickness fringes were observed. In either powder, no
clear evidence of domains was observed. The absence of domains, however, remains
inconclusive, because in general, the largest particles did not allow for transmission of the
electron beam, and in addition, specific particle orientations would be required in order to
observe the domains. Literature supports the existence of polydomain BaTi03 particles at
particle sizes > ~ 0.5 fim, with results being somewhat ambiguous below ~ 0.2 fim
(Wakino, 1994).
_
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
<»
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 7.18. TEM images o f hydrothermal powders at 200,000 X magnification: (a)
bright field image of BT-8 powder, (b) dark field image of BT—8 powder, (c) bright field
image o f BT—16 powder, and (d) dark field image of B T -16 powder.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
181
7.2.4 Composite Microstructural Characterization Using SEM
Composite samples fabricated using the various powders in Table 6.1 were prepared at
50 volume percent powder loadings, using polypropylene as the matrix material, as
outlined in Chapter VI, section 6.2. To distinguish between these various samples the
following identification scheme will be employed: the large particle size B aTi03 (TAM, 1.0 pm) powder-polym er composites will be designated as TAMC; the fine particle size
BaTi03 (BT—8, ~ 0.2 pm) powder-polymer composites will be designated as B T-8C ; and
the very fine particle size B aTi03 (BT—16, ~ 30 nm) powder-polymer composites will be
designated as BT—16C. By taking the density o f B aT i03 to be 6.02 g/cc, and the density o f
polypropylene to be 0.90 g/cc, and noting that the composite contains 50 volume percent of
each component, it is a simple matter to show that the expected density of the composite is
3.45 g/cc. The range of geometric densities determined for each o f the various composite
samples used throughout this study are reported in Table 7.20 along with their percent
expected values.
Table 7.20 Measured------Densities
—
-- ----------of BaTiQ3 Powder—Polymer Composites
Sample
Density
Percent Expected Density
(g/cc)
TAMC
3 .41-3.44
-99
B T -8C
3.32-3.38
-97
B T -16C
3.13-3.23
-92
Scanning electron microscopy (SEM) was used to examine the composites in order to
characterize the connectivities of the powder phase. Figures 7.19, 7.20, and 7.21 show
the microstructure o f the TAMC, BT—8C and BT—16C, composites, respectively. In
general, the particles were well distributed throughout the matrix. Because of the fine size
of the BT-16 powder, the degree of particle clustering or agglomeration could not be
determined.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 7.19. Microstructure o f TAMC composite.
Figure 7.20. Microstructure o f BT-8C composite.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
183
Figure 7.21. Microstructure of BT—16C.
7.2.5 Dielectric Spectroscopy o f Composite B aTi03 Powder—Polymer Composites
The dielectric properties (dielectric constant and dissipation factor) of the various
BaTi03 polymer composites were measured through microwave frequencies at room
temperature using the techniques described in Chapter VI. As in the case with the ceramic
samples, when possible, at least five or more samples were measured at each frequency.
This is the case for all composite samples unless otherwise indicated. From these
measurements conducted on multiple samples at each frequency, an average was taken.
This is the reported value in all subsequent Tables and figures. The error presented along
with the average measured values was determined from the maximum and minimum
measured values from any set of samples at a particular frequency.
As previously discussed, polypropylene was selected as the matrix material for the
composites because of its reported low loss properties in the microwave frequency range.
To confirm this, right cylindrical rods o f polypropylene o f various dimensions were
fabricated and used to measure their dielectric properties using the post resonance
technique. Table 7.21 summarizes the measured parameters and calculated dielectric
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
184
properties. From Table 7.21, it can be seen that the K is virtually frequency independent,
and the loss remains on the order ~ 1C4.
Diameter
(cm)
2.521
1.875
1.272
Table 7.21 Microwave Properties of Polypropylene
Length
K
fo
Qu
(cm)
(GHz)
1.426
9.682
2.274
1889
0.973
13.807
2421
2.267
0.474
25.671
808
2.270
tan5
0.0005
0.0004
0.0012
Low frequency (100 Hz to 10 MHz) dielectric measurements were conducted on the
composite samples, using the method discussed in chapter VI, section 6.5. Error analysis
was conducted in the same way as described in previous sections. Table 7.22 shows the
room temperature data measured from the TAMC samples in the frequency range of 100 Hz
to 10 MHz.
Table 7.22 Room Temperature Low Frequency Dielectric Properties o f TAMC
V
Frequency
+K
-K
^average
+tan5
-tanS
100 Hz
38.432
0.5764
0.7686
0.014
0.001230
0.00110
10 kHz
37.166
0.5946
0.4088
0.0027
0.000214
0.000213
100 kHz
36.933
0.5909
0.4062
0.0027
0.000218
0.000218
400 kHz
36.812
0.5889
0.4049
0.0032
0.000255
0.000255
1 MHz
36.750
0.5880
0.4042
0.0032
0.000252
0.000252
10 MHz
36.389
0.5822
0.4002
0.0016
0.000128
0.000128
Similarly, a lumped impedance technique was used to obtain dielectric data in the
frequency range between 50 MHz and 1 GHz using the technique outlined in Chapter VI,
section 6.6. Table 7.23 illustrates typical raw data for a representative TAMC sample
having a diameter o f 3.03 mm and a thickness o f 0.818 mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
185
Table 7.23 Representative Raw Data and Calculated K and tan8 From Lumped Impedance
Measurements Conducted on TAMC
Frequency
K
0
tanS
|f(0 )|
(MHz)
(degrees)
50
0.9989
-5.21
0.0123
36.05
100
0.9974
-10.36
36.36
0.0147
200
0.9941
-20.45
36.17
0.0172
300
0.9907
-30.25
36.13
0.0188
400
0.9871
-39.63
36.12
0.0207
500
0.9842
-48.50
36.13
0.0216
600
0.9804
-56.87
36.19
0.0240
700
0.9775
-64.66
36.25
0.0256
800
0.9753
-71.86
36.32
0.0268
900
0.9733
-78.54
36.43
0.0281
1000
0.9724
-84.68
36.53
0.0286
Table 7.23 illustrates another fundamental problem associated with this technique in the
measurement of low—ATmaterials having relatively low losses. Recall that from Chapter
in, section 3.2.2, the effect of the fringing field was to increase the measured K ' while
K " remained unaffected, that is, the effect of the fringing field would be to increase K and
lower tanS from its true value. It is believed that this effect is affecting the measured
dielectric properties of the composite samples at frequencies generally above 400 MHz,
consequently, lumped impedance results above this frequency were not used. Note that the
fringing field effect is expected to be significant only with the low K composite samples,
because in the higher K dielectrics (ceramics), the electric field lines tend to concentrate
more in the dielectric, thus reducing the effect. Table 7.24 summarizes the results obtained
from lumped impedance measurements on five different TAMC samples of dimensions
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
186
characterized by diameters between 3.23 and 3.02 mm, and thicknesses between 1.04 and
0.800 mm.
Table 7.24 Room Temperature Lumped Impedance Measurements on TAMC
F
Frequency
+K
-K
average
+tan6
-tanS
^average
(MHz)
50
36.388
0.65200
0.80000
0.0092500 0.0030730 0.0028000
100
36.288
0.68240
0.72460
0.012700
0.0020200
0.0029700
200
36.098
0.68200
0.70000
0.016290
0.0042200
0.0033400
300
35.872
0.71060
0.69640
0.018388
0.0056120
0.0035280
400
35.835
0.73160
0.68540
0.020311
0.0085000
0.0044000
A cavity perturbation technique was used for the measurement o f dielectric properties at
selected frequencies above 1 GHz. A TMj,,, resonating cavity was employed as outlined in
Chapter VI, section 6.6, and as discussed earlier in this chapter. A summary of the
measured parameters, and calculated dielectric parameters is provided in Tables 7.25
through 7.28.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
R eproduced with permission
of the copyright ow ner.
Table 7.25 Measured Parameters and Calculated Dielectric Properties of TAMC Using Cavity Perturbation at 1.5 GHz
Sample
*c1,2average
y'1, 2average
(
Vt
Vo
/,
(cm)
(cm)
(cm3)
(cm3)
(GHz)
1.599
0.01836
291.77
1.5059
796
<2,
IL
prohibited without p e rm issio n .
A f //0
K
tanS
-37.7
0.004
31.62
0.0456
30.85
0.0474
(dB)
1
0.1030
(cm)
0.1115
2
0.1080
0.1090
1.596
0.01879
291.77
1.5059
788
t
oo
o
Further reproduction
/„ = 1.5116 GHz, <20 = 1030-1087
0.004
3
0.1095
0.1040
1.597
0.01819
291.77
1.5064
791
-38.0
0.003
29.13
0.0512
4
0.0840
0.0855
1.600
0.01149
291.77
1.5077
814
-37.6
0.003
33.26
0.0636
5
0.1005
0.1120
1.598
0.01799
291.77
1.5061
1030
-35.3
0.004
32.06
0.0422
00
-J
R eproduced with permission of the copyright ow ner.
Table 7.26 Measured Parameters and Calculated Dielectric Properties of TAMC Using Cavity Perturbation at 3 GHz
Further reproduction
f 0 = 3.0227 GHz, 0 O= 1320
Sample
^1,2 average
(cm)
l. 2average
£
v,
^0
/,
IL
(dB)
4 f//0
K
tan5
(cm)
(cm1)
(cm3)
(GHz)
1.598
0.01867
73.9
2.9850
627
-44.7
0.027
27.25
0.0341
prohibited without p e rm issio n .
1
0.1025
(cm)
0.1140
2
0.0840
0.0855
1.598
0.01148
73.9
2.9987
831
-41.6
0.029
28.99
0.0274
3
0.1035
0.1090
1.599
0.01804
73.9
2.9876
745
-41.9
0.024
27.14
0.0247
4
0.1075
0.1090
1.595
0.01869
73.9
2.9870
627
-43.7
0.031
26.73
0.0349
5
0.1005
0.1120
1.599
0.01799
73.9
2.9862
632
-43.5
0.025
28.22
0.0336
00
00
R eproduced with permission
of the copyright ow ner.
Table 7.27 Measured Parameters and Calculated Dielectric Properties of TAMC Using Cavity Perturbation at 5.6 GHz
Further reproduction
f{) = 5.5942 GHz, Q0 = 275
Sample
^1,2 average
s'
1, 2average
t
(cm)
<2,
Vo
/,
(cm)
x V‘
101*
(cm’)
(cm3)
(GHz)
1.252
7.814
17.6
5.4811
234
IL
A ///0
K
tan5
-10.7
0.021
28.07
0.0276
(dB)
prohibited without p e rm issio n .
1
0.0790
(cm)
0.0790
2
0.0790
0.0800
1.249
7.894
17.6
5.4849
230
-10.8
0.019
26.89
0.0305
3
0.0770
0.0780
1.252
7.519
17.6
5.4934
247
-10.1
0.018
25.93
0.0243
4
0.0770
0.0780
1.251
7.513
17.6
5.4878
240
-10.5
0.019
27.43
0.0263
5
0.0790
0.0790
1.251
7.809
17.6
5.4866
240
-10.5
0.019
26.72
0.0260
00
vO
190
From the data shown in Tables 7.25 through 7.27, the average dielectric properties along
with their associated error are shown in Table 7.28.
Table 7.28 Room Temperature Cavity Perturbation Measurements on TAMC
jr
Frequency
+K
-K
^■average
-tanS
+tan5
^average
(GHz)
1.50
31.39
1.87
0.0500
2.26
0.00779
0.01361
2.99
27.67
1.32
0.936
0.0309
0.003933
0.00625
5.50
27.00
1.06
1.075
0.0269
0.00358
0.00263
Using the data shown in Tables 7.22, 7.24 and 7.28, the spectrum of dielectric properties
for the TAMC sample was plotted as a function of the logarithmic of frequency. These
results are shown in figure 7.22.
In figure 7.22 the dielectric constant of TAMC decreases from its low frequency value
o f ~ 37 measured at 10 kHz to its high frequency value o f ~ 27 measured at ~ 5 GHz. The
dielectric loss gradually increases from its low frequency value of 0.003 measured at 10
kHz, traverses a maximum of - 0.05 at 1.5 GHz, and rapidly decreases to ~ 0.027 at ~ 5
GHz. The A K is thus ~ 10, and by defining the frequency at which tan5 peaks as the
relaxation frequency, f r, for T A M C ,/r = 1.5 GHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
191
Dielectric Loss, tanS
CN
©
©
—
o
OO
o
o
"S'
VO
o
o
©
o
CN
o
o
o
ON
VO
N
X
m oo
o
00
CO
C4
©
cn
o
CM
X ‘JUH1SU0 3 OUJOOpiQ
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 7.22. Spectrum of dielectric properties measured from TAMC composite.
OO
192
Low frequency, lumped impedance and cavity perturbation measurements were also
conducted on B T -8C samples. Table 7.29 shows the room temperature data measured
from the BT-8C samples in the frequency range of 100 Hz to 10 MHz.
Table 7.29 Room Temperature Low Frequency Dielectric Properties of BT-8C
F
Frequency
+K
-K
^average
+tanS
-tan8
tanSaveragc
100 Hz
39.20
0.5660
0.8310
0.02414
0.002160
0.001980
1 kHz
38.23
0.6000
0.7460
0.01403
0.000680
0.000680
10 kHz
37.63
0.5646
0.7280
0.00892
0.000176
0.000104
100 kHz
37.13
0.5510
0.6960
0.00688
0.000120
0.000180
400 kHz
36.89
0.5480
0.6890
0.00684
6.0000e-05
0.000140
1 MHz
36.76
0.5440
0.6800
0.00634
0.000160
0.000240
10 MHz
36.27
0.5420
0.6650
0.00530
0.000700
0.000500
Similarly, a lumped impedance technique was used to obtain dielectric data in the
frequency range between 50 MHz and 400 MHz. Table 7.30 summarizes the results
obtained from lumped impedance measurements on five or more different BT—8C samples
of dimensions characterized by diameters between 2.98 and 3.06 mm, and thicknesses
between 0.634 and 0.890 mm.
Table 7.30 Room Temperature Lumped Impedance Measurements on BT-8C
F
+K
Frequency
-K
^average
-tanS
+tan5
tan8average
(MHz)
35.94
1.190
0.00128
0.00032
50
2.184
0.00512
100
35.86
1.140
2.177
0.00792
0.00148
0.00122
200
35.76
1.170
2.147
0.01041
0.00279
0.00171
300
35.75
1.190
2.142
0.01141
0.00359
0.00231
400
35.75
1.233
2.122
0.01231
0.00419
0.00301
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
193
A cavity perturbation technique was used for the measurement o f dielectric properties at
selected frequencies above 1 GHz. A TM qI0 resonating cavity was employed as outlined in
Chapter VI, section 6.6, and as discussed earlier in this chapter. Tables 7.31 through 7.34
summarize the measured results.
_
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 7.31 Measured Parameters and Calculated Dielectric Properties of BT-8C Using Cavity Perturbation at 1.5 GHz
/ 0 = 1.5133 GHz, 00 = 881-886
Sample
C
J1
,2average
c'1,2 average
£
v,
V0
/,
(cm)
(cm)
(cm3)
(cm3)
(GHz)
1.597
0.01425
293.32
1.5089
773
<2,
IL
A ///0
K
tanS
-38.5
0.0029
33.06
0.0268
(dB)
1
0.097
(cm)
0.0920
2
0.0960
0.0960
1.598
0.01473
293.32
1.5086
727
-39.0
0.0031
34.65
0.0375
3
0.0940
0.0955
1.596
0.01433
293.32
1.5087
783
-38.1
0.0031
35.01
0.0226
4
0.0970
0.0935
1.599
0.01473
293.32
1.5087
785
-38.1
0.0039
36.01
0.02087
5
0.0950
0.0960
1.598
0.01457
293.32
1.5084
828
-37.8
0.0033
37.14
0.0117
6
0.0960
0.0940
1.598
0.01442
293.32
1.5084
857
-37.4
0.0033
36.95
0.0049
R eproduced with permission
of the copyright ow ner.
Table 7.32 Measured Parameters and Calculated Dielectric Properties of BT-8C Using Cavity Perturbation at 3 GHz
/„ = 3.0229 GHz, Q 0 = 1100-1800
Further reproduction
Sample
■^1.2average
c'
1, 2average
I
v,
Vo
/.
(cm)
(cm)
(cm1)
(cm1)
(GHz)
1.598
0.01404
72.7
2.9878
978
0.
IL
A T //o
K
tanS
-39.0
0.012
33.46
0.0196
(dB)
prohibited without p erm issro n .
1
0.6945
(cm)
0.0935
2
0.0930
0.0965
1.598
0.01427
72.7
2.9879
848
-40.8
0.012
33.18
0.0218
3
0.0945
0.0965
1.596
0.01455
72.7
2.9902
870
-40.2
0.011
30.12
0.0133
4
0.0955
0.0960
1.599
0.01466
72.7
2.9865
888
-40.2
0.012
33.19
0.0176
5
0.0960
0.0935
1.598
0.01434
72.7
2.9860
881
-40.4
0.012
34.38
0.0232
6
0.0930
0.0955
1.598
0.01419
72.7
2.9906
807
-41.0
0.011
30.77
0.0218
R eproduced with permission
of the copyright ow ner. Further reproduction
Table 7.33 Measured Parameters and Calculated Dielectric Properties of BT-8C Using Cavity Perturbation at 5.6 GHz
/ 0 = 5.5942 GHz, Q 0 = 275
Sample
c1,2average
s '1, 2average
t
V.v
V0
/.
(cm)
(cm)
(cm-1)
(cm1)
(GHz)
1.249
0.01092
16.2
5.37598
204
Q^
IL
A ///0
K
tanS
-11.3
0.039
31.73
0.0163
(dB)
prohibited without p e rm issio n .
1
0.9350
(cm)
0.0935
2
0.0925
0.0935
1.249
0.01080
16.2
5.37369
194
-11.7
0.039
32.45
0.0192
3
0.0935
0.0945
1.248
0.01120
16.2
5.38912
174
-12.1
0.036
29.14
0.0286
4
0.0900
0.0920
1.248
0.01039
16.2
5.39271
200
-11.1
0.036
30.79
0.0189
5
0.0930
0.0930
1.248
0.01079
16.2
5.38543
211
-10.8
0.034
30.73
0.0148
6
0.0925
0.0945
1.249
0.01092
16.2
5.38300
190
-11.5
0.038
30.74
0.0213
7
0.0940
0.0925
1.248
0.01085
16.2
5.37382
201
-11.4
0.039
32.24
0.0170
VO
On
197
From the data shown in Tables 7.31 through 7.33, the average dielectric properties along
with their associated error are shown in Table 7.34.
Table 7.34 Room Temperature Cavity Perturbation Measurements on BT-8C
Frequency
+K
-K
^average
+tan8
-tanS
tanSavcrage
(GHz)
1.50
35.46
1.67
2.40
0.02073
0.01673
0.01583
2.99
32.51
1.86
2.40
0.01954
0.00361
0.00629
5.50
31.11
1.33
1.98
0.01945
0.00912
0.00460
Using the data shown in Tables 7.29, 7.30 and 7.34, the spectrum of dielectric properties
for the BT-8C sample was plotted as a function of the logarithmic of frequency. These
results are shown in figure 7.23.
In figure 7.23 the dielectric constant o f BT-8C decreases from its low frequency value
of ~ 37 measured at 10 kHz to its high frequency value o f ~ 31 measured at ~ 5 GHz. At
the highest measured frequency, it appears that the dielectric constant has not yet saturated
out to its high frequency value. The dielectric loss gradually increases from its low
frequency value o f 0.009 measured at 10 kHz, appears to pass through a broad maximum
which peaks at ~ 0.02 and is centered at 3.16 GHz.
Finally, low frequency and post resonance measurements were conducted on BT-16C
samples. The post resonance technique was selected for these measurements because the
trend in the microwave losses o f TAMC and BT-8C samples suggested that the losses
would be sufficiently low to use this technique. This technique offers the highest accuracy
of the methods employed in this work. Table 7.35 shows the room temperature data
measured from the BT-16C samples in the frequency range of 100 Hz to 10 MHz.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
198
Table 7.35 Room Temperature Low Frequency Dielectric Properties o f BT-16C
IT
Frequency
+K
-K
average
+tan5
-tan8
tanSaverage
100 Hz
34.07
0.0284
0.00560
0.00340
0.6200 0.3600
200 Hz
33.74
0.6200
0.3400
0.0236
0.00430
0.00220
400 Hz
33.44
0.6200
0.3500
0.0199
0.00318
0.00162
1 kHz
33.09
0.6220
0.3380
0.0159
0.00212
0.00180
4kH z
32.82
0.5480
0.4120
0.0118
0.00110
0.00065
10 kHz
32.52
0.6160
0.3440
0.0104
0.00072
0.00051
100 kHz
32.18
0.5800
0.2400
0.0077
0.00026
0.00042
400 kHz
32.30
0.6060
0.4140
0.0070
0.00063
0.00061
1 MHz
32.17
0.6130
0.4170
0.0068
2.9600e-05
0.00069
10 MHz
31.62
0.5660
0.4040
0.0061
0.00045
0.00203
Using the composite material BT-16C, several dielectric rod resonators were fabricated
o f various dimensions for dielectric property measurements in the microwave frequency
range, using the post resonance technique. Table 7.36 summarizes these measurements.
Table 7.36 Room Temperature Post Resonance Measurements on BT—16C
Diameter
Length
K
fo
Qu
tan5
(cm)
(cm)
(GHz)
2.543
1.610
2.675
29.6
0.0091
110
0.0124
1.911
1.132
81
30.3
3.619
1.280
3.283
5.212
66
31.5
0.0151
The spectrum of dielectric properties for BT-16C is shown in figure 7.24. In figure
7.24 it can be seen that the dielectric constant decreases from its low frequency value of ~
33 measured at 10 kHz to 29.5 measured at 2.6 GHz. This gradual decrease in K is also
accompanied by a low, flat loss of ~ 0.007 through the same frequency range. Above 2.6
GHz, both K and tanS begin to increase, suggesting the onset, as in the case of the fine
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199
grain ceramic, of a resonance phenomenon. This resonance may be attributed to the
existence of a certain percentage of particles in the particle size distribution behaving as
single domain particles. If this is the case, the resonant frequency could then be related to
the width of the resonating particle. Unfortunately, using the techniques outlined in this
work, higher frequency measurements of the necessary accuracy could not be done. It is
clear that the apparent resonant frequency o f the BT-16C sample is higher than that of the
FGBT ceramic. This is expected because of the smaller particle size o f the BT—16 particles
compared to that of the grain size of FGBT, as well as the fact that in the case o f the
ceramic, it is likely that the grain resonances were coupled, which led to a lowering o f the
resonance frequency.
Figure 7.25 compares the high frequency spectra of the various composite samples.
Error bars are not shown for clarity. At the lowest measured frequencies (not shown in
figure 7.25), interfacial polarization increased with the surface area of the powders. These
effects were manifested mainly in the dielectric losses, o f which BT—16C had the highest,
and TAM C had the lowest. At higher frequencies, beginning at 1 MHz and shown in
figure 7.25, the effects of interfacial polarization have been eliminated. The dielectric
properties of the TAMC and BT-8C composites are virtually equivalent up to ~ 100 MHz.
Above this frequency, the tanS of the TAMC material undergoes a more rapid increase,
with increasing frequency. The BT-8C material also begins to show some marginal
increase in its tanS as well. The K of the TAMC material starts to dramatically decrease
above ~ 400 MHz, while the K of the BT-8C material remains relatively flat until - 1 .5
GHz, above which, it begins to decrease more rapidly. The BT-16C material yielded the
lowest K measured at 10 MHz. The K of the BT-16C material exhibited a very gradual
decrease with increasing frequency, up to - 2.5 GHz; above this frequency the onset of
resonant behavior is apparent.
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200
Dielectric Loss, tan5
VO
OO
fN
r©
©
r-
ON
cs
00
©
©
VO
©
©
VO
OO
cs
m
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VO X
oo
o
«n
eo
cn
CN
©
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m
X
‘JOBJSUO3 O lU O O piQ
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 7.23. Spectrum of dielectric properties measured from BT-8C composite.
oo
201
Dielectric Loss, tanS
cs
©
o
On
O
CO
NO
o
O
©
On
NO SI
SC
00
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OO
CO
CN
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«o
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CN
X'umsuoz) oujospiQ
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 7.24. Spectrum of dielectric properties measured from BT-16C composite.
00
B T -iC
0.14
0.12
35
TAMC
v>
§
o
O
BT-
30
0.08
TAMC
prohibited without p e rm issio n .
0.06
3
n
a
Dielectric Loss
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40
BT-8C
Oj
0.04
25
BT-16C
0.02
20
J_
6
6.5
7
7.5
8
8.5
9.5
log/ (Hz)
Figure 7.25. Comparison of dielectric spectra of various composite samples.
K)
O
N>
The dielectric losses of both the TAMC and B T-8C materials pass through a peak which
appears to broaden and shift to a higher frequency with decreasing particle size. The tanS
o f the BT—16C material appears to be approaching a peak at some higher frequency. The
broadening of the peak is attributed to the decrease in domain wall width, as the domain
size decreases with particle size. In the case of the TAMC and BT-8C materials, where the
loss appears to go through a maximum, a domain width, d, may be calculated from f r using
equation (4.31). Domain widths of 0.48 and 0.23 p m were calculated for TAM C and B T 8C, respectively. This suggests that the TAM particles are indeed polydomain, with each
particle possessing a few number of twins. In the case of the B T-8 particles, it is possible
that a significant percentage of the particles are larger than the mean particle size o f 0.19 pm
(see figure 7.16 (b)), whereby these larger particles may possess a few number of twins of
this size, or perhaps even a little larger. A small percentage of particles towards the low
end o f the distribution may be behaving as single domain particles. The absence of
resonance character in the dielectric spectrum suggest that there exist a distribution of
domain sizes, however, the fact that the calculated domain width is approaching the
predominate particle size may suggest the at this size, ~ 0.2 pm, a critical size is being
approached, whereby the particles are behaving as single domain particles.
A calculated Ps value may be obtained for the TAM particles from the measured AK of
the TAMC material. Because the relaxation spectra o f the other composite materials are
incomplete, this analysis is not possible. In order to determine Ps, the AK value o f the
TAM particles must be determined. For this, a dielectric mixing model was used to
determine the K of the TAM particles from the measured K of the TAMC composite. The
model employed was developed by simulating the mixing between two different materials
using finite element analysis, fully accounting for dielectric polarization and infringing of
the electric flux due to the discontinuities at the material boundary regions. The model
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204
assumes no special tendency to form chains and/or clusters (Wakino, eta l., 1993). The
equation governing the dielectric constant of the mixture, Km, is given by (7.2),
Km = exp
ln{V^AT1(V'~Vi)>+ V2K i2v'~Vo>}
(7.2)
v t - v 2
where, Km, AT, and K2 are the dielectric constants o f the mixture, material 1 (TAM), and
material 2 (polypropylene), respectively, and V, and V2 are defined as the volume fraction
of material 1 and material 2, respectively. The value V0 is defined as the critical volume
fraction at which, the equation (7.2) intercepts that predicted by the logarithmic model. The
value o f VQwas demonstrated to be just below 0.60 by studying two component mixtures
containing C a T i0 3 particles of 1—3 |im in size. For this work, VQ= 0.60 will be used.
Using equation (7.2), with V0 = 0.60, the dielectric constant of the TAM particles was
determined, and is plotted in figure 7.26. Figure 7.26 shows that the K decreases from
1766 at 1 M Hz to 731 at 5.5 GHz, with a AK = 1035. Using this value in equation (4.31),
Ps was calculated as 25 pC/cm2, very close to the value known for BaTiOj.
2000
1800
1600
co
U
cj
•c
!
o
a
1000
800
600
6
6.5
7
7.5
8
8.5
9
9.5
log/ (Hz)
Figure 7.26. Dielectric spectrum o f TAM particles using mixing rule.
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10
205
7.3 Domain Size Correlation
In those cases where it was possible to measure through the relaxation frequency, for
both ceramic and composite samples, a domain width was calculated. A summary of the
calculated domain widths is presented in Table 7.37.
Sample
CGBT
TAMC
BT-8C
Table 7.37 Summary o f Calculated Domain Widths
calculated domain
grain/particle size
fr
width
(Hm)
(Um)
14.4
771MHz
0.98
1.33
1.50 GHz
0.48
0.23
0.19
3.16 GHz
Although the grains in the ceramic are believed to be subject to a stress field different
from that experienced by the particles in a polymer matrix, it is nevertheless, of interest to
examine the domain size dependence on grain/particle size using both the ceramic and
composite data. Figure 7.27 shows the calculated domain width as a function o f average
grain/particle size. The domain width followed a power law of the form,
d = Cdng lp,
(7.3)
where d is domain width, C is a constant equal to 0.413, n is a constant equal to 0.334,
and dglp is the size of the grain/particle. The power law dependence has been previously
observed, however, the constant n determined here, differs from that o f ~ 0.5 supported by
literature. The difference is attributed to the fact that previous studies focused on ceramics,
which necessarily exhibited anomalous stress induced domain structures. In this work, the
smallest particles were embedded in a polymer matrix, thereby modifying the stress field to
some extent, thus, it is believed, the domain states were intrinsic to particle size, and not
due to high internal stress relief. From equation (7.3), the size at which the domain width
and grain/particle size are equivalent was determined to be 0.265 Jim.
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2
1.5
0.5
0
0.1
10
Average Grain/Particle Size ((im)
Figure 7.27 Correlation between domain width and grain/particle size.
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100
CHAPTER VIII
SUMMARY AND FUTURE WORK
8.1 Summary
In this work, polycrystalline ceramic ferroelectric BaTiOj and BaTiO, powder-polymer
matrix composites were prepared, and investigated in order to understand the effects of
grain size and particle size on the crystallographic deformation, domain structure, low
frequency dielectric properties and high frequency (microwave) dielectric properties. High
frequency measurements were conducted to identify and correlate relaxation mechanisms
with the material characterization results. The composite samples were employed to
decouple resonances between adjacent grains as well as reduce the three dimensional
clamping, experienced by grains in a ceramic. It was felt that this would lead to a domain
structure intrinsic to particle size, and not dominated by stress relief mechanisms.
Specifically, by changing grain size and particle size, the domain states o f the crystallites
were modulated, which subsequently influenced the high frequency dielectric properties of
these materials.
Using high purity starting commercial powders, ceramics possessing average grain
sizes o f 14.4 (coarse grain), 2.14 (small grain), and 0.26 p m (fine grain) were processed
using conventional sintering and hot pressing techniques. Microstructural evaluations
revealed clear domain patterns in both the coarse grain ceramic and small grain ceramic.
X-ray diffraction studies on the ceramics showed a marked decrease in tetragonality for the
fine grain ceramic. The psuedocubic structure was explained by the presence of unrelieved
stress persisting through the grain, resulting in a suppression of the tetragonal distortion
characterized by ferroelectric B aT i03. The presence of this high unrelieved stress was
attributed to the absence o f domain twinning in the majority of the grains.
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208
Temperature dependent low frequency measurements were used to determine the
transition parameters. The coarse grain and small grain ceramics exhibited distinct peaks in
the dielectric constant associated with the ferroelectric to paraelectric transition, as well as
the lower temperature transitions. The small grain ceramics possessed the highest K at all
measured temperatures below Tc. This was attributed to the increased domain twinning
occurring in the grains o f this size due to a two dimensional stress relief mechanism. The
fine grain ceramic did exhibit a clear peak associated with the ferroelectric to paraelectric
transition, however, the lower temperature transitions were much more diffuse. Again,
this was attributed to stress related crystallographic constraints. In addition, ferroelectric
hysteretic properties showed that the small grain ceramic had the highest remanent
polarization while the fine grain material had the smallest.
Several microwave dielectric measurement techniques were discussed and used for the
determination of dielectric properties above 10 MHz. The various techniques employed
include the lumped impedance technique, the cavity perturbation technique and the dielectric
post resonance technique. Each technique had associated with it specific limitations. The
lumped impedance technique exhibited calibration related problems when used to measure
high K ceramics, generally above 500 MHz. Low K materials were more susceptible to
fringing field effects. The cavity perturbation technique, in theory, has no real limitation,
however, in practice, requisite sample sizes may be unattainable in order to maintain the
validity of the approximations made in the derivation o f this technique. Finally, the
dielectric post resonance method demands low loss (high Q) materials so that resonant
peaks are observed and correctly identified.
Using the various microwave measurement techniques, the dielectric spectra of the
various ceramic samples were measured. All three samples exhibited evidence o f relaxation
or resonance in their dielectric spectrum. The coarse grain ceramic exhibited Debye-like
relaxation with a AK of ~ 1700 and a relaxation frequency,/r, of 771 MHz. Using the
5
i
i
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209
relaxation models, a domain size o f 0.98 (am and a Ps of 32 flC/cm2 were calculated from
the measured relaxation parameters. The calculated domain size correlated well with the
observed microstructure. The high value o f Ps was attributed to the high value of AK. At a
grain size o f 14 pm , its suggested that the domain twinning pattern is entering into a strong
dependence on grain size, resulting in some marginal increase in the low frequency K.
The small grain ceramic material exhibited the highest K and loss tangents compared to
the other ceramic samples. For this reasons, its dielectric properties could only be
determined up to 200 MHz. Efforts to measure its dielectric properties using the cavity
perturbation technique failed, indicative of its persistent high K and lossy behavior well
into the microwave frequency range. Measurements up to 200 MHz, however, clearly
revealed the onset o f the relaxation phenomenon, which appeared to occur at ~ 100 kHz.
The dielectric spectrum of the fine grain ceramic exhibited distinct resonant character.
The K of ~ 2000 remained relatively flat up to ~ 100 MHz, above which it clearly began to
increase. Associated with the increase in K was a corresponding increase in tan8, which
appeared to peak to ~ 0.3, at ~ 1.6 GHz. The resonant behavior o f the dielectric spectrum
was attributed to the piezoelectric resonance of single domain grains. By describing the
equivalent circuit of a grain around its resonant frequency, predicted resonant frequencies
ranging between 5 .1 and 8.5 GHz were calculated from the available grain size data. These
values seemed slightly high when compared to the measured spectrum, and was explained
by the fact that the length of the resonant units was increased through intergrain coupling,
thereby lowering the effective resonant frequency.
In general, the relaxation properties were explained by the observed domain structures
in the ceramics. Compared to the coarse grain ceramic, the high frequency properties of the
small grain ceramic showed a more lossy, broader relaxation, indicative of an increase in
domain wall density and decrease in domain wall diameter. Upon decreasing grain size to
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210
~ 0.26 pm, the relaxation took on a distinctively resonant character, defined by lower
losses and an apparent shift in the relaxation frequency. This was attributed to the
existence of single domain grains, which exhibited coupled resonances.
Again, using high purity starting commercial powders, powder-polymer
(polypropylene) composites possessing average particle sizes o f 1.33 p m , 0.19 p m , and
66 nm, as determined from PSD and BET measurements, were processed. X-ray
diffraction studies on the powders showed a systematic decrease in tetragonality with
decreasing particle size. TEM microstructural characterization did not reveal the presence
o f domain twinning in the hydrothermal powders, however, those results were at best,
inconclusive due to the limitations of this characterization technique. It was proposed that
some of the particles towards the high end of the PSD for the B T-8 powders could be
undergoing domain twinning, but this could neither be confirmed nor disproved. TEM
observations also showed that the BT-16 particles were highly agglomerated, possessing
very irregular shapes.
Using the various microwave measurement techniques, the dielectric spectra of the
various composite samples were measured. All three samples exhibited evidence of
relaxation or resonance in their dielectric spectrum. The TAM (1.33 pm) composite
exhibited Debye-like relaxation with a AK of ~ 10 and a relaxation frequency,/,., of 1.5
GHz. The BT-8 (0.19 p m ) composite also exhibited relaxation in its dielectric properties.
The relaxation, however, was clearly shifted with a broadening of the loss peak, compared
to that of the TAM composite. No clear resonant character was discerned in the relaxation
spectrum of the BT—8 composite. The relaxation frequency was determined to be ~ 3.2
GHz. Using the relaxation models, a domain size o f 0.48 and 0.23 p m was calculated for
the TAM composite and BT-8 composite, respectively. The fact that the particle size of the
BT-8 powder appears to be coinciding with the domain size suggests that ~ 0.2 p m may be
close to a critical size at which quasi-stress free particles begin behaving as single domain
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211
particles. The absence of resonant character in the relaxation spectrum for the BT-8
composite was not distressing, because of the distribution of particle sizes present in the
powder. The larger particles present (possibly as large as a micron), may have exhibited
domain twinning, leading to a larger distribution of domain sizes. The BT—16 composite
exhibited a clearly lower K at all frequencies, as well as the lowest high frequency losses
up to ~ 5 GHz. At 2.5 GHz the K and loss began to show an increase, possibly due to the
onset of some resonant phenomenon. A possible explanation is the existence of single
domain particles behaving as independent piezoelectric resonators. Unfortunately due to
the limitations of the measurement techniques in this work, higher frequency properties
could not be measured to confirm the resonance character of the spectrum.
In general, the relaxation spectra broadened, and shifted to higher frequencies with
decreasing particle size. The smallest particles (~ 66 nm) appeared to show the onset of
resonant character at the highest measured frequencies. With decreasing particle size, the
high frequency losses also decreased. This was explained by the fact that the particles
exhibited a more systematic variation in domain configuration with size, due to a significant
decrease of the stress field experienced by the particles compared to that of grains in a
ceramic. The low frequency properties of the TAM and BT-8 composites were virtually
identical; the BT-16 composite clearly had a lower K. By using a dielectric mixing model,
the K of the ferroelectric particles was calculated. From the measured AK of the TAM
particles, a Ps of 25 pC/cm2 was calculated. This value agrees very well with the expected
value for BaTiO,.
Finally, the domain size dependence on grain/particle size was examined by using the
calculated domain widths of the coarse grain ceramic, the TAM composite and the BT-8
composite. The trend in domain size with decreasing grain/particle size was fitted to a
power law. The power law was used to determine the size at which the particle size and
domain width are equivalent. The size was determined to be 0.265 pm.
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212
8.2 Future Work
In this work, the highest frequency measurements were conducted to ju st below 6 GHz.
In some cases, the full relaxation spectra was not obtained, even at 6 GHz. In those cases
where the spectra was truncated, higher frequency measurements are necessary. It is
possible to extend the range o f measurements for particularly the composite materials by
designing and fabricating additional TMo,0 cavities that resonate at higher frequencies. A
potential problem, however, is the degradation of the 0 -fac to r of the cavity with increasing
frequency. The cavity Qs used in this study, decreased from ~ 1300 at 1.5 GHz to ~ 300 at
5.6 GHz. High Q cavities are necessary for accurate loss measurements.
Another future goal should be the development of a broadband swept frequency
technique of sufficient accuracy for the measurement of ferroelectric materials. One
technique that was attempted as part of this study was the use of filled X -band
waveguides, however, the errors associated with this technique led to excessive scatter in
measured data, and consequently were not reported in the results of this work. The coaxial
transmission line offers the appeal of no cutoff frequency for propagating TEM modes.
This offers the potential of extending swept frequency measurements from as low as 50
MHz to well into the microwave frequency range. A method whereby the cross section of
the guide is field with the test material could be explored (analogous to the X—band
measurements), or as was done as part of this study, the technique involving the
termination of the line with the sample could be further investigated. The concerns,
however, are that at higher frequencies higher order modes begin to propagate which may
lead to cross mode coupling errors. The ratio of inner conductor radius to outer conductor
radius can be tailored to some degree to control higher mode propagation, but there is a
practical limit. The calibration and fringing field problems must also be overcome, as was
illustrated by this study.
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213
Temperature dependent measurements should be conducted as well. These
measurements would reveal the effects of temperature on the relaxation frequency.
Measurements o f this type were attempted using the lumped impedance method, however,
there was no way to perform a calibration at elevated temperatures. Consequently, errors
due to changes in conductivity and thermal expansion of the test fixture could not be fully
accounted for, which again led to gross errors in the data. High temperature
measurements, above the Tc, should reveal the effects of no domains or domain walls on
the high frequency spectra. Cryogenic measurements would reveal the effects o f a clamped
domain state on the high frequency spectra.
Another way of modulating the domain state is through poling. Poling grows certain
domains at the expense of others. It is expected that the coalescence of the domain
structure, resulting in larger average domain sizes, and fewer domain walls, should result
in the lowering o f the relaxation frequency with an accompanying decrease in K and
dielectric losses. Studies of this type should be undertaken as well.
As discussed in Chapter I, ferroelectrics as phase shifter materials present an attractive
alternative for currently used materials. Their main limitation, however, were their high
losses. This work presents at least one strategy to lower these high losses. Another
attractive characteristic o f the ferroelectric composite materials is that they would offer
lower impedance mismatch problems when employed in a real device. For this reason, the
voltage tunability of the composite materials should be investigated at high frequency.
BaTiO, was chosen for this study because it is a well behaved, first order prototype
ferroelectric, that has been well studied. Other ferroelectric compositions (for example,
barium strontium titanate), more attractive for device implementation should be studied
using the same approach. Other connectivities could be investigated as well. This work
employed the use o f essentially 0-3 composites. Other connectivites, for example, 1—3
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214
connectivites should yield higher K composites. Finally, their voltage tunablility should
also be studied at both low and high frequencies.
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I
!
VITA
Mark Patrick McNeal was bom on December 23, 1964 in Allentown Pennsylvania, to
Mr. and Mrs. Alphonso McNeal. He graduated from the Pennsylvania State University in
May 1990, with a B.S. in Ceramic Science and Engineering. He then received a M.S.
degree in Solid State Science in May 1994, also at the Pennsylvania State University. He
was a recipient of an International Society for Hybrid Microelectronics (ISHM) Educational
Foundation Grant in 1991. In January 1994 he entered the doctoral program in Materials,
at the Pennsylvania State University. He currently resides with his wife Kelley, in
Marlborough Massachusetts.
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