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Microwave dielectric properties measurements by various resonance techniques on the potential substrate materials for high critical temperature superconductor thin films

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8 0 0 /5 2 1 -0 6 0 0
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M icrow ave d ielectric p r o p e r tie s m ea su rem en ts b y variou s
reso n a n ce tech n iq u es o n th e p o te n tia l su b str a te m a teria ls for
h ig h T c su p erco n d u cto r th in film s
S h een , J y h , P h .D .
The Pennsylvania State University, 1993
UMI
300 N. Zeeb Rd.
Ann Aibor, MI 48106
The Pennsylvania State University
The Graduate School
M IC R O W A V E D IE LEC TR IC PR O PER TIES M EA SU R EM EN TS BY V A RIO U S
R E SO N A N C E TE C H N IQ U E S ON T H E PO T EN TIA L SU B ST R A T E
M A T ER IA L S FO R H IG H Tc SU PE R C O N D U C TO R TH IN FILM S
A Thesis in
Electrical Engineering
by
Jyh Sheen
Subm itted in Partial Fulfillm ent
o f the Requirem ents
for the Degree o f
D octor o f Philosophy
D ecem ber 1993
W e approve the thesis o f Jyh Sheen.
D ate o f Signature
- ''
■
9
■■ c
r
.
.
f -g
- "
1
_________
L. Eric C ross
Evan Pugh Professor o f Electrical Engineering
Thesis A dviser
Co-C hair o f C om m ittee
A m ar S. B halla
Senior Scientist and Professor o f M aterials
Thesis A dviser
C o-C hair o f C om m ittee
F rancis T. S. Yu
E van Pugh Professor o f Electrical Engineering
I/i 5/? 3
Kenji Uchir
Professor o f Electrical Engineering
? A
y
?
3.______
ran A. Carpentm^
A ssociate Professor of Electrical Engineering
t i n
Larry C. BurU
Professor o f Electrical Engineering
H ead o f the D epartm ent o f Electrical and
C om puter Engineering
h
i
A BSTRA CT
A system atic study and com parison on the measurem ent techniques o f dielectric
constant and dielectric loss at microwave frequencies has been undertaken. A m odification
to the Itoh and R udokas M odel for com putations o f the resonant frequency, and losses o f
the T E ois m ode o f the parallel plate dielectric resonator have been suggested. That
m odification has shown accurate prediction on resonant frequency and reasonable
com putations on conductor loss and radiation loss.
T he dielectric properties o f suggested new substrate materials for high Tc
superconductor thin films were measured at m icrowave frequencies using different
m icrow ave resonance techniques at 300 °K and 77 °K. Those m aterials have shown
adequate dielectric properties as the substrate materials for high Tc superconductor thin
films for the applications at m icrowave frequencies. The accuracy o f different m icrowave
m easurem ent techniques on dielectric constant and dielectric loss m easurem ents is also
proved. T he relationship betw een microwave dielectric properties and frequency is also
studied.
The dielectric properties o f the com posite samples m ade by the m ixed powders of
B a(M gi/ 3 T a 2 / 3 ) 0 3 and polyethylene have been measured at X -band frequencies. The
com parison, has also been made betw een the m easured results with two w ell-know n
m ixture rules, and with a new reported mixture rule. The adequacy o f using that pow der
mixture m ethod to predict the dielectric properties o f pure Ba(M g i /3 Ta 2 /3 )C>3 at m icrow ave
frequencies is discussed.
iv
TABLE OF CONTENTS
LIST O F FIGU R ES
viii
LIST O F TA B LES
xii
ACKNOW LEDGEM ENTS
.......................................................................................
xiv
C hapter 1. IN TR O D U C TIO N
.......................................................................................
1
1.1. Criteria for Choice o f Substrate M aterial ..............................................................
3
1.1.1. Consideration o f Dielectric C o n stan t........................................................
3
1.1.2. Loss C onsiderations.....................................................................................
6
1.1.3. Lattice M atching
.......................................................................................
9
1.1.4. Therm al Expansion P ro p erties..................................................................
11
1.1.5. O ther R equirem ents.......................................................................................
11
1.2. Literature R eview on High Tc Superconductor Substrate M aterials...............
11
1.2.1. Characteristics of Substrate M aterials.......................................................
13
1.2.2. Sum m ary
15
1.3. Frequency D ependence o f M icrow ave Dielectric P ro p erties............................
19
1.4. Purpose o f R esearch
.......................................................................................
20
C hapter 2. M IC R O W A V E M EA SU R EM EN T T E C H N IQ U E S................................
22
2.1. D ielectric R esonance Techniques............................................................................
23
2.1.1. Post R esonance T echnique.........................................................................
24
2.1.2. C ylinder C avity Resonance T echnique.....................................................
34
2.1.3. W aveguide Reflection Resonance Technique ........................................
37
2.2. Parallel Plate Resonance T echnique........................................................................
40
2.2.1. Introduction
.....................................................................................
42
V
2.2.2. Field Expressions .......................................................................................
44
2.2.3. Com putations and E xperim ents................................................................
48
2.2.4. M icrowave M easurem ents Using the Parallel Plate Dielectric
R esonator
53
2.3. O ther Param eters o f D ielectric Resonance Techniques......................................
56
2.3.1. C alculation o f the Factor "A ”..................................................................
56
2.3.2. Calculations o f Conductor and Radiation Losses...................................
62
2.3.3. Error A nalysis for Dielectric C onstant M easurem ents.........................
66
2.3.4. Error A nalysis for Q Factor M easurem ents...........................................
70
2.3.5. Liquid N itrogen Tem perature M easurem ents.........................................
75
2.3.6. Sum m ary of Dielectric Resonance T e c h n iq u e s ....................................
75
2.4. Perturbation Techniques
.......................................................................................
77
2.4.1. Rectangular C a v ity .......................................................................................
81
2.4.2. Cylindrical Cavity .......................................................................................
85
2.4.3. Re-entrant Cavity
.......................................................................................
89
2.5. Transm ission Techniques .......................................................................................
92
2.5.1. D istributed Transm ission Technique........................................................
92
2.5.2. Lum ped Im pedance Technique.........................................
97
2.6. C onclusion o f M icrow ave M easurem ent T echniques.........................................
100
C hapter 3. SE L E C T IO N A N D PR EPA R A TIO N O F N EW SU B ST R A T E
M A T E R IA L S
.......................................................................................
103
3.1. Selection o f New Substrate M aterials...................................................................
103
3.1.1. A (B iB 2 ) 0 3 Fam ily
103
3 .1.1.1. B a(M gi/3Ta2/3)03.......................................................................
104
3.1.1.2. S r(A li/ 2 Nb 1/ 2 ) 0 3 , Sr(A l[/ 2 T a 1/2 ) 0 3 , S r(G ai/ 2 T a i/ 2 ) 0
106
3
...
vi
3.1.1.3.
0 .7
Sr(A l]/ 2 X i/ 2 )O 3 : 0.3LaA 10 3 and
0 .7
Sr(A Ii/ 2 X i/ 2 )O 3 :
0.3L aA 10 3 (X= Nb, T a )...........................................................
3.1.1.4. B asic Properties o f A (B iB 2 )0
3.1.2. L M A n O
19
3
F a m ily ..........................
F am ily.......................................................................................
3.1.3. K M F 3 Fam ily
106
107
109
.......................................................................................
110
.....................................................................................
113
........................................................................................
113
3.2.2. Laser Heated Pedestal G row th...................................................................
113
C hapter 4. E X PE R IM E N T A L R ESU L TS A N D D IS C U S S IO N S ............................
118
4.1. M easurem ent Results o f U sed M aterials...............................................................
119
4.2. M easurem ent Results o f the O xide F a m ily ..........................................................
120
4.2.1. M easurem ent o f Conductivity o f M etal S h ie ld s....................................
121
4.2.2. M easured R esults o f Ceram ic Sam ples...................................................
122
4.2.3. Frequency D ependence o f Dielectric P ro p erties....................................
129
4.2.4. M easured R esults o f Single Crystal Fiber S am ples..............................
135
4.3. M easurem ent R esults o f the K M F 3 F a m ily .........................................................
136
4.4. Sum m ary
136
3.2. Sam ple Preparation
3.2.1. Ceram ic S amples
C hapter 5. E STIM A TIO N O F TH E D IELEC TR IC PR O PER TIES OF
B a(M g ,/ 3 Ta2/3)03 B Y PO W D ER M IX IN G M E T H O D ...................
5.1. Introduction
5.2. Sam ple Preparation
139
........................................................................................
5.3. M ixture R ules
5.4. R esults and D iscussions
5.5. Sum m ary
139
140
140
.......................................................................................
141
149
vii
C hapter
6
. C O N C L U SIO N S A N D FU TU R E W O R K ....................................................
151
6.1. C onclusions
151
6 .1 .1 .
M easurem ent T echniques............................................................................
151
. 1.2. Characteristics of the Potential Substrate M aterials...............................
152
6
6.2. Future W ork
REFEREN CES
155
157
viii
LIST OF FIGURES
Figure
1.1.
M icrostrip transm ission lin e .......................................................................................
4
1.2.
C oplanar m icrostrip transm ission lin e ......................................................................
8
1.3.
C rystal structure o f the YBa 2 Cu 3 0 7 _x ......................................................................
10
2.1.
M easurem ent setup.
........................................................................................
25
2.2.
Post resonance technique..............................................................................................
26
2.3.
Post resonator m ode chart o f a dielectric sample with dielectric constant 23,
diam eter 11.48 m m , and thickness 3.33m m ...........................................................
27
2.4.
Typical resonant curve o f the transm ission signal.................................................
28
2.5.
R elationship o f resonant frequency and sample dim ensions o f the post
resonance technique.
.......................................................................................
33
2.6.
C ylinder cavity resonance technique.........................................................................
35
2.7.
M ode chart o f the cylinder cavity resonator with support thickness
6
.1 mm
and dielectric sample with dielectric constant 23, diam eter 11.48 mm, and
thickness 3.33 m m .
36
2.8.
W aveguide reflection resonance technique.............................................................
38
2.9.
M ode chart of the reflection dielectric resonator with a C band cavity and
support thickness 7.6 mm. The dielectric sample has dielectric constant 23,
diam eter 11.48 m m , and thickness 3.33 m m .........................................................
39
Resonant curve o f the waveguide reflection resonance tech n iq u e......................
41
2 .1 1. Parallel plate resonance technique..............................................................................
43
2.10.
2.12.
N um erical and experim ental results o f the resonant frequency o f the parallel
plate resonance technique............................................................................................
49
ix
2.13. Calculated Q c and Qr by the Itoh & Rudokas M odel and the m odified
m ethod o f the parallel plate resonance technique..................................................
51
2.14. Com puted and experimental results o f quality factors of the parallel plate
resonance technique.
.......................................................................................
52
2.15. M ode chart o f the parallel plate resonator with Li=L-2 = 4.6 mm and dielectric
sample w ith dielectric constant 23, diameter 11.48 mm, and thickness 3.33
mm.
54
2.16. Relationship o f resonant frequency and sample dimensions o f the parallel
plate resonance technique w ith L i= L 2 = 5m m .......................................................
55
2.17. Calculations o f the "A" factor by equations (2.13) and (2 .1 8 )............................
58
2.18. Relationship o f factor "A" and the sample dim ensions of the post resonance
technique.
59
2.19. Relationship o f factor "A" and sam ple dim ensions o f the parallel plate
resonance technique.
2 .2 0
.......................................................................................
60
. Calculated Q c of the parallel plate resonance technique by using equations
(2.12a) and (2.19).
2.21.
Quality factor due to the conductor loss o f the parallel plate resonance
technique.
2.22.
68
N orm alized measurement errors o f the dielectric constant due to the error on
dim ension measurements o f the parallel plate resonance technique..................
2.25.
65
N orm alized measurem ent errors o f the dielectric constant due to the error on
dim ension m easurem ents o f the post resonance technique.................................
2.24.
64
Quality factor due to the radiation loss o f the parallel plate resonance
technique.
2.23.
63
R elationship o f B/A and the sam ple dim ensions o f the post resonance
69
X
technique.
.......................................................................................
72
N orm alized measurem ent error o f quality factor due to the error on pow er
level m easurem ent of the waveguide reflection resonance technique for
......................................................................................................
76
C avity perturbation technique using rectangular c a v ity ........................................
79
C avity perturbation technique using cylindrical c a v ity .........................................
80
M ode chart o f the X band rectangular cavity w ith length 13.5 c m ....................
84
C avity perturbation technique using re-entrant cavity..........................................
90
D istributed transmission technique..........................................................................
93
Setup o f device under test o f distributed transm ission technique.......................
96
D ube transm ission technique.....................................................................................
98
L um ped im pedance technique...................................................................................
99
Processing procedure of the ceram ic preform ........................................................
114
Schem atic diagram o f the laser heated pedestal grow th station..........................
117
f5min<<' !•
Q uality factors o f perovskite oxide family m easured by the parallel plate
resonance technique.
.......................................................................................
131
D ielectric constants o f perovskite oxide family measured by the parallel plate
.......................................................................................
132
R elation o f fxQ and frequency o f perovskite oxide fa m ily .................................
134
resonance technique.
D ielectric constants o f B M T -PE composites m easured by the S I 1 and S21
.......................................................................................
144
C om parison o f experim ental d ata and theoretical m ixture rules........................
145
technique.
Estim ation o f BM T dielectric constant. Data w ere m easured by the S 11 and
S21 technique.
.......................................................................................
Estim ation o f BM T dielectric constant. Data w ere m easured by the cavity
146
xi
perturbation technique.
5.5.
147
Estim ation o f B M T dielectric constant. Data w ere m easured by the post
resonance technique.
5.6.
.......................................................................................
.......................................................................................
148
Dielectric loss o f BM T-PE com posites m easured by the S 11 and S21
technique.
150
xii
LIST OF TABLES
Selection criteria for substrate materials for high Tc superconductor thin films.
12
D ielectric properties o f substrate materials for high Tc superconductor films.
16
Som e basic properties o f substrate m aterials for high Tc superconductor
.......................................................................................
18
Comparison betw een four dielectric resonance techniques.................................
78
film s.
C om parison o f m easured quality factors (Qm) and the theoretical quality
factors (Qth) o f two rectangular cavities used for the cavity perturbation
technique (TE m odes).......................................................................... .......................
86
Sum m ary o f microw ave dielectric m easurem ent techniques.............................
101
Properties o f com plex perovskite com pounds at m icrow ave freq u en cies......
105
Properties o f K M F 3 fa m ily .......................................................................................
108
Ceram ic processing conditions.................................................................................
109
Dielectric properties o f potential substrate materials o f H TSC at 10 k H z.......
111
Some basic properties o f some perovskite oxide materials o f potential
substrate o f HTSC.
.......................................................................................
112
Processing procedure o f the ceram ic p re fo rm .......................................................
115
Dielectric properties o f AI 2 O 3 single crystal fiber and LaAlC >3 single crystal.
119
Param eters o f the shield conductivity m easurem ents..........................................
121
M icrowave dielectric properties at room tem perature o f the oxide family o f
the potential substrate materials for HTSC thin films at ceram ic form.
Samples were m easured by dielectric resonance techniques
.......................
M icrowave dielectric properties at room tem perature o f the oxide family o f
123
xiii
the potential substrate m aterials for H TSC thin film s at ceram ic form.
Sam ples w ere m easured by the cavity perturbation tech n iq u e ..................
4.5.
D ielectric constants o f the L M A i 1O 19 fam ily in ceram ic form m easured by
the cavity perturbation technique......................................................................
4.6.
125
127
M icrow ave dielectric properties at liquid N 2 tem perature o f the oxide fam ily
o f the potential substrate m aterials for H T S C thin film s at ceram ic form .
S am ples w ere m easured by cylinder cavity resonance techniques..........
4.7.
127
D ielectric constants o f single crystal fibers m easured by the cavity
perturbation technique...........................................................................................
135
4.8.
D ielectric properties o f the K M F 3 fam ily........................................................
137
5.1.
D ielectric properties o f B M T -PE com posite sa m p le s.................................
142
xiv
ACKNOW LEDGEM ENTS
I w ould like to express my deepest thanks to Dr. A. S. B halla for being m y thesis
adviser and for his help and enlightening discussions. I w ould also like to express my
deepest gratitude to Dr. L. E. Cross, my other thesis adviser, for giving me the chance to
be his student and for his help and guidance during the past four years.
I w ish to thank Dr. F. T. S. Yu not only for serving on my com m ittee, but also for
giving his sincere advice on my future career. I would also like to thank Dr. L. A.
C arpenter for supplying useful inform ation and suggestions for my research. I w ish also to
express m y gratitude to Dr. K. Uchino for serving on my com m ittee, and for taking the
tim e to read my thesis.
I w ish to express m y special thanks to Dr. Ruyan G uo fo r her generous help during
the past three years and for supplying very useful data in my thesis. I would also like to
express my gratitude to Dr. C. J. Peng for helping me prepare the com posite sam ples. I
w ish also to thank Dr. D. Kajfez, Electrical Engineering D epartm ent, U niversity o f
M ississippi, for supplying the com puter program used in my m easurem ents. I w ould also
like to thank Dr. S. E rdie, Dr. P. Ravindranathan, and Dr. U. Selvaraj for supplying
sam ples for my m easurem ents. I m ust also thank Dr. R ustum Roy, Dr. F. W . A inger, Dr.
E . C. Subbarao, and D r, G. Harshe for useful com m ents and discussions.
Finally, I w ish to thank my family. I m ust express my deepest gratitude to m y wife,
Y u-C hen Liu, for her constant support and understanding, and for giving me the m ost
invaluable gift-- m y son, Alan. My parents m ade many sacrifices for m e during their lives.
A ll my deepest thanks should be given to them.
1
Chapter 1
IN TR O D U C TIO N
T he occurrence o f superconductivity in the Y Ba 2 Cu 3 0 7 .x (YBCO) opened up the
possibility o f superconducting devices w hich can operate with liquid nitrogen as the
refrigerant. Currently, there is increasing interest in the applications o f thin film
superconductors. O ne o f the major potential applications o f the low loss property o f the
high Tc superconductor (HTSC) thin film s is for devices operating at m icrow ave
frequencies. The H TSC thin films have an im portant role in m icroelectronics, to make
possible low dispersion, high speed, dense superconducting interconnects. A m ajor
obstacle to the above applications is the ability to deposit the HTSC as thin films on low
loss and low dielectric constant substrate materials which are com patible w ith Si or other
sem iconductors. For certain applications, a loss tangent o f 10 4 to 10*5 is required for the
substrate materials. A nother potential application of HTSC thin films is the propagation of
extrem ely short electrical pulses w ithout any resistive loss as long as the bandw idth o f the
electrical pulses does not exceed the energy gap o f the superconductor which is about
10
terahertz (THz) for YBCO. However, due to the surface roughness and m icrostructure of
the superconducting film s, the resistivity o f the superconductor is too great to obtain low
enough surface resistance at very high frequencies. The choice o f substrate m aterials is
very im portant to ensure high quality film s with surface resistance several orders below
that o f the copper. A t this time, no fully satisfactory substrate material has been found for
the depositions o f Y B C O thin films. It is w orthw hile to search for a new substrate
material.
This thesis is dedicated to the measurements o f the dielectric constant and the dielectric
loss (or quality factor) at m icrow ave frequencies o f the new HTSC substrate materials. In
this chapter, the criteria for choosing the substrate material are described in section
1.1.
Section 1.2 gives a general review o f the properties o f m aterials used as the substrates for
Y BCO films. Section 1.3 gives a simple description of the dielectric dispersion theory at
m icrow ave frequencies. A sum m ary o f the research purpose o f this thesis w ill be given in
section 1.4.
B efore the m easurem ent results o f substrate materials are given and discussed, the
different m icrow ave m easurem ent techniques will be discussed in chapter 2. A general
review o f m icrow ave m easurem ent techniques w ill be presented. However, techniques
adequate for m easuring the dielectric properties o f substrate materials for HTSC thin films
will be the m ain focus. A newly modified microwave resonant technique will also be
presented in this chapter. Chapter 3 describes the basic properties (other than the
m icrow ave dielectric properties) o f new suggested materials to be used as substrates o f
H TSC thin films. The m easured results o f the m icrowave dielectric properties o f those
m aterials will be given in chapter 4. The adequacy of those substrate materials for
m icrow ave applications will be discussed. T he m easured results o f som e used substrate
materials for H TSC thin films will also be given in this chapter. C hapter 5 studies the
dielectric properties o f the com posite samples m ade by m ixing the powders o f
B a(M gi/ 3 T a 2 / 3 ) 0 3 and polyethylene. The novel B a(M g[/ 3 Ta 2 / 3 )C>3 m aterial, has been
suggested as the substrate for H TSC thin films; its dielectric properties are estim ated by
using the pow der m ixing method. The conclusion and future w ork suggestions will be
given in chapter 6 .
3
1.1. C riteria for Choice o f Substrate M aterial
The criteria for choosing substrate material depend upon the requirem ents o f
applications. The dielectric constant influences the propagation speed and the package
dim ensions. Since the conductor loss o f the superconductor is dim inished, the role o f the
dielectric loss becom es more important. In addition to the dielectric properties (the main
focus in this thesis), lattice, thermal expansion, chemical, and m echanical properties of the
substrate are also critical for the qualities o f the grown films.
1.1.1. Consideration of the Dielectric Constant
T he requirem ents for the dielectric properties o f substrate materials at m icrow ave
frequencies for superconducting film depend on the characteristics o f desired device. The
propagating characteristics o f chip-to-chip interconnecting lines often limit the overall
perform ance o f today’s electronic designs. By using superconducting thin film as the
interconnect, the low resistance can reduce the rise time and dramatically increase the
operation speed o f high speed systems (K w on et al., 1987). The role o f the delay due to the
transm ission time becom es m ore important. The phase velocity o f the typical m icrostrip
line in part (a) Figure 1.1 using superconductors as the conductor strips can be expressed
as (K autz, 1979):
Vp = V7 (1 +T C0V
(‘ D
w here c is the free space light speed, e' is the dielectric constant value o f substrate material
(er = e' + E ", er is the relative com plex permittivity), t is the conductor thickness, and X is
the penetration depth which is the order o f 0.1 p m for YBCO. W ith conductor thickness
4
S trip conductor
^
J£~
d
^
e'
(a)
vwwwvv
(b)
F igu re 1.1. C onfigurations o f (a) m icrostrip (b) shielded strip transm ission lines.
5
close to or larger than 1 pm , the velocity is a function o f dielectric constant only. Since the
phase velocity o f signal decreases as the dielectric constants o f substrate materials increase,
low dielectric constant values are desired. In addition, the characteristic im pedance for the
shielded stripline configuration in part (b) o f Figure 1.1 which allows m ultilayer
interconnect processing can be expressed as (Liao, 1990, p. 489),
w ith W /d »
94.15
1
Ve
W + 044
d
0.35 and t «
d. For a certain characteristic im pedance, a low er dielectric
constant means a sm aller layer thicloiess and higher packaging density. The noise from the
crosstalk o f adjacent traces can also be reduced by using thinner layers (M ohideen et al.,
1988). Furtherm ore, the radiation efficiencies o f m icrostrip dipole antennas are higher for
substrate m aterials w ith sm aller dielectric constant values (Rutledge et ah, 1983). In
general, although a low er dielectric constant is desired, substrate materials with dielectric
constants lower than
20
are acceptable.
From the other point o f view, in som e m icrow ave devices (delay line, for exam ple;
A nderson et al., 1983), a lower dielectric constant m eans a low er w avelength reduction
factor. The size w ill be increased. The dielectric constant is expected not to be low er than
10. For U H F circuits, the substrate area can be reduced by using a dielectric constant o f 20
to 25. However, for higher frequencies, the required dielectric constant o f substrate is
sm aller (Talvacchio et al., 1991). G enerally, a substrate dielectric constant around 20 is
adequate.
6
1.1.2. Loss Considerations
The total attenuation o f the microstrip line can be expressed as:
a = a c+ a d + a r
(1.3)
w here etc, otd, and otr are the attenuations associated with conductor, dielectric, and
radiation losses respectively. In general, for frequencies below 30 G H z, if there is no
discontinuity on the line, the radiation loss is negligible. The attenuations of the microstrip
line in part (a) o f Figure 1.1 caused by the conductor loss for W > d (sim ilar conditions on
the follow ing discussions can be applied to W < d) and dielectric loss can be calculated by
(Liao, 1990, p. 479),
8 .6 8 6
«c
=
a d =
- vZ
R,
0w
"
27.3 * (e - 1 ) tan 5
-----------------------------
ee( e ' - l ) X g
dB /m
( 1 .4)
dB /m
(1.5)
w here Rs is the surface resistance o f the conductor, Zq is the characteristic im pedance of
the stripline, Ee is the effective dielectric constant o f the microstrip line, tan5 is the loss
tangent o f the substrate material (tan 6 = e ’Ve' = 1/Q, Q is the quality factor), and
w avelength in the dielectrics. The Rs is proportional to
( f r e q u e n c y ) 1^
is the
and (frequency ) 2 for
norm al conductor and superconductor, respectively (Lyons et al., 1990).
F o r the microstrip line in part (a) o f Figure 1.1 with a typical characteristic impedance
Z G= 50 £2, linew idth w = 100 pm , substrate dielectric constant e'= 10 (Ee= 6 .8 , w /d = 1.1),
and using copper as the conductor, the attenuation caused by the conductor loss at room
tem perature on a typical nonsem iconductor substrate is about 0.52 dB per w avelength for
frequency at 10 GHz. By using high Tc superconductor as the conductor to reduce the
7
surface resistance, the conductor loss can be reduced dramatically. For a superconductor
stripline with surface resistance
10
times lower than that o f copper at liquid nitrogen
tem perature, the attenuation by conductor loss can be reduced to the order o f 0.019 dB/Xg
at 10 GHz. By keeping the the same conductor loss as the copper stripline at room
tem perature, from equation (1.4), the dim ensions o f the superconductor m icrostrip line are
only one-thirtieth (w = 3 to 4 pm ) o f those o f the copper stripline. From equation (1.5), the
attenuation caused by the dielectric loss can be kept much lower than the conductor loss
w ith loss tangent tanS less than 10*3. However, it was proven that, for the application o f
sm all superconducting dipole antenna with significantly higher efficiencies, a loss tangent
o f less than
10*4
is necessary for any dielectric m aterials in the transm ission lines and
support structures for the antenna. For dipole length less than 0.1 w avelength, 10' 5 w ill be
desirable (D inger et al., 1990). The attenuation due to conductor loss and stripline
dim ensions can be further reduced by im proving the film quality to low er surface
resistance.
A nother potential application o f YBCO films is the propagation o f extrem e short
electrical pulses w ithout any resistive loss as long as the bandw idth o f the pulse does not
exceed the energy gap o f superconductor. In the case o f YBCO that w orks to about 10 TH z
(N uss et al., 1989). A pair o f coplanar m icrostrip lines as shown in Figure 1.2 was studied
for propagating picosecond electric pulses (Nuss et al.,1989; G rischkow sky et al., 1987).
T he radiation loss occurs because the propagation velocity o f the pulse is faster than the
phase velocity in the underlying dielectric (G rischkow sky et al., 1987). The radiation loss is
calculated by (Rutledge et al., 1983),
2.32 w V l + 1/e’
af
3
Xg K (k) K ’(k)
N p/m
P
(1.6)
Strip conductors
X ~
H*-
> K > |< ------w
s
w
d
Figure 1.2. C oplanar m icrostrip transm ission line.
9
w here K and K' are standard tabulated elliptic integrals, and k= s/w. For frequencies higher
than 100 GHz, the losses are dom inated by the radiation loss. The upper lim it o f tan 5 is
5 x l0 *4 at 500 G H z for a 20 pm wide line (Nuss et al., 1989). By using frequency at 10
G H z as judgem ent, the requirem ent for loss tangent may be as low as at the order of 10*4
to
1 0 ’5
because the loss tangent m ay increase with increasing frequency at the microwave
frequency region, although this trend may not be adequate from the order o f several
gigahertz to the order o f hundreds of gigahertz. From equation (1.6), the radiation loss can
be reduced by decreasing the stripline width. However, in a sim ilar situation w ith the single
m icrostrip line, reducing the substrate thickness will increase conductor loss as strip
linew idth decreases, especially at high frequencies because the surface resistance of
superconductor is proportional to the square o f frequency. In order to decrease the design
dim ensions and keep the conductor loss low, the requirem ent for surface resistance o f
superconductor film at 10 G H z may be as low as 100 or 1,000 times low er than that o f
copper at liquid N 2 temperature. Such a low surface resistance requires very good substrate
material characteristics - lattice matching, thermal expansion coefficient m atching etc.,
w hich will be discussed next.
1.1.3. Lattice M atching
The Y BCO high tem perature superconductor is oxygen deficient oxides with a
perovskite crystal structure. That structure consists o f near planar arrays o f copper and
oxygen atom s spread throughout, with layers o f barium or yttrium atom s. T he unit cell of
the high tem perature superconductor is show n in Figure 1.3. Dim ensions o f the unit cell
are a = 0.3820 nm , b = 0.3892 nm, and c = 1.1688 nm (c/3 = 0.3896 nm ). In order to have
the films with required orientation, substrate materials should have lattice constant as close
to 0.38 to 0.39 nm as possible.
10
o o
•co
V
.
Figure 1.3. C rystal structure o f the YBa 2 C u 3 0 7 _x superconductor.
11
1.1.4. Therm al Expansion Property
M atching o f thermal expansion coefficients between the thin films and the substrate
m aterials is another important factor for the quality o f thin films. It is desired that the
therm al expansion properties can be matched at room, deposition, annealing, and
application tem peratures for proper film conditions. The YBCO has thermal expansion
coefficient a a= 14 ppmy°C, otb= 9 ppm /°C , and ctc= 19 ppm /°C for a, b, and c three axial
directions respectively. For grow ing films w ith c-axis perpendicular to the film surface
plane, the desired thermal expansion coefficient of substrate is 9 to 14 ppm/°C.
1.1.5. O ther R equirem ents
In addition to the electrical and structural properties, one m ust consider other criteria
for choosing the substrate m aterials. The substrates should be single dom ain; no tw inning
due to phase transitions in tem perature range between substrate grow th and application
tem peratures. The chemical properties should be stable at deposition tem perature, and
annealing tem perature. M echanically, the substrate should be strong for scratch resistance
and no cleavage in the surface plane. The crystals can be grow n, are easy to process, and
cost little. O ther factors, for exam ple, availability o f different geometries (fibers or planar
structures), required orientation for epitaxy films, and low defect density should also be
considered. A sum m ary o f the criteria for choosing substrate m aterials is listed in Table
1. 1.
1.2. Literature Review on H igh Tc Superconductor Substrate M aterials
M any m aterials have been used as the substrates for Y BCO thin films. The first
substrate identified for achieving highly-oriented growth o f HTSC film was SrTi 0 3 , the
12
Table 1.1.
Selection criteria for substrate materials for high Tc superconductor thin films
for m icrowave applications.
Properties
Requirem ents
Electrical
1. Low dielectric constant: e'< 25 (depends on the applications);
2. Low dielectric loss: tan5< lxlO*3 at liquid nitrogen temperature.
Structural
1. Good lattice matching;
2. Therm al expansion matching.
C hem ical
1. Stable at deposition and annealing temperatures;
2. Stable w ith tim e (no change at interface);
3. Stable to the ambient.
M echanical
1. N o cleavage property in the surface plane o f interface;
2. Single dom ain (no twinning).
O ther factors
1. Availability o f different geometries (fiber o r planar structures);
2. Required orientation for deposition;
3. Low defect density.
13
m ost used substrate material for YBCO. A fter that, M gO, YSZ (yttria stabilized zirconia),
and sapphire (AI 2 O 3 ) crystals were used. From the consideration o f m icroelectronics
devices, silicon single crystal was used, as well as quartz crystal. Later, LaG a 0 3 , N dG a 0 3 ,
L aA 1 0 3 , K Ta 0 3 , and Y bFe 0 3 etc. were reported being used as the substrates o f high Tc
films. In the following, the characteristics for individually used substrate m aterials are
discussed so that the selection criteria for materials used as the substrates o f Y BCO thin
film s are more clear.
1.2,1. Characteristics o f Substrate Materials
SrTiQ 3 Strontium titanate is a cubic perovskite material with a good lattice matching
(a= 0.3905 nm ) with the YBCO. SrTi 0 3 was the first substrate identified that prom oted
highly oriented grow th o f HTSC films. The very high dielectric constant o f SrTi 0 3 which
increases from 200-3 0 0 at room tem perature to about 2,000 at liquid nitrogen tem perature
(K obayashi et al., 1989) severely limits in the operation speed for devices based on
SrTi 0 3 - The high dielectric loss o f strontium titanate also limits its applications.
Y SZ T he fluoride type structure with oxygen ion spacings 0.257 nm are not suitable
for epitaxy o f Y B CO w here the oxygen spacings on the copper-oxygen are 0.273 nm ,
which leads to polycrystal film (Tietz et al., 1989). Chemical reactivity is low. The
dielectric loss is too high for most microwave applications (Thoro et al., 1973).
M eO The M gO has face-centered cubic structure. Its low dielectric constant (~ 10)
and low loss (K onaka et al., 1991) are very attractive for m icrow ave applications.
C hem ically, because o f its low reactivity, M gO is a good substrate. However, like YSZ,
the distance betw een oxygen ions (0.298 nm) has very large m ism atch to that o f YBCO.
A s w ith Y SZ, Y BCO films grown on M gO show ed polyciystalline also (Tietz et al.,
1989).
14
AI2 O 3 Both single crystal and polycrystal o f AI2 O 3 have been used for the substrates
o f HTSC. It has a low cost, is available in large untw inned wafers, and is physically strong
and robust. Sapphire also shows low dielectric constant and very low dielectric loss
(K onaka et al., 1991). However, the reactivity o f alum inum show s poor film quality (Naito
et al., 1987). The lattice m ism atch w ith YBCO is great. YSZ or C e0 2 buffer layers are
needed for film deposition. The anisotropy o f its dielectric constant and its thermal
expansion m ism atch w ith YBCO are tw o more disadvantages.
Si & SiP2 The annealing process forms an interlayer of copper silicide. A buffer
layer is needed.
LaGaQ3 & LaAlQ3 Lanthanum gallate and lanthanum alum inate are the only two
substrate crystals grow n specifically for high Tc oxide superconductors. L aG a 0 3 has a
perovskite structure w ith orthorhom bic sym m etry, a = 0.5526 nm , b = 0.5473 nm, and c 0.7767 nm . The lattice constants are a = c = 0.3888 nm, and b = 0.3884 nm as a
pseudocubic cell. It is very close thermal expansion m atch to YBCO. LaG a 0 3 was found
to have a moderate dielectric constant (~ 25) and loss tangent 2x1 O'4 at liquid nitrogen
tem perature (Sandstrom et al., 1988). Som e tw inning exists on lanthanum gallate.
LaA 1 0 3 is a rhom bohedral perovskite w ith lattice constant a = 0.3792 nm. It has a low
dielectric constant and a low loss. Dielectric constant was reported to be 23 and the loss
tangent w as 5 .8 x l0 ’4 at 300 °K and 8 .3 x l0 ‘5 at 77 °K (Sim on et al., 1988). T he main
disadvantage o f L aA 1 0 3 is that it suffers from twin boundary moving at the grow th/anneal
tem peratures resulting in poor surface morphologies.
N dG a 0 3 is another m em ber o f the perovskite family w ith a cubic structure at room
tem perature w ith a = 0.3837 nm. High quality c- oriented film s have been grow n on this
substrate (Koren et al., 1989). The N d G a 0 3 has a dielectric constant 23 and a loss tangent
IxlO -3 at 300 °K and 3xl0*4 at 77 °K. H ow ever, the tw inning also exists in the material.
15
K Ta03 It is cubic perovskite with a = 0.399 nm and has also been used as a substrate
for deposition o f Y B C O w ith a or c axes perpendicular to the substrate (Feensta et al.,
1989). KTa 0 3 has dielectric constant 2,000, w hich is not suitable for microwave
applications. The loss tangent value is 2xlO -3 at 80°K (B elokopytov et al., 1990).
LiN b03 It is rhom bohedral and has the advantages o f low cost and ready availability.
L iN b 0 3 has a dielectric constant 39 and dielectric loss 4x1 O'2 (Xi et al., 1983). YBCO
film s have been grow n on LiN b 0 3 , with film to substrate orientation in all three
crystallographic directions (Hohler et ah, 1989).
Alkaline Earth Fluorides M F 2 ( M = M g, Ca, Sr, and B a) are cubic with lattice
param eters o f 0.546 nm to 0.620 nm. They have low dielectric constants (4.7 to 7.3) and
loss tangents < 10-4 at 4 °K (Bystrov et ah, 1986). YBCO grow n on those substrates
yielded only polycrystalline films. Reaction w as detected betw een CaF 2 and the film. A
buffer layer is required.
M iscellaneous B eO , GaAs, InP, Y bFe 0
3
, and other m aterials have been used as the
substrates o f Y B CO films. Little success has been reported w ith those substrates.
1.2.2. Sum m ary
The properties o f the above discussed substrate materials are sum m arized in Tables
1.2 and 1.3. From the above analysis, there w as no substrate m aterial with fully
satisfactory characteristics yet discovered. T he search for new substrate materials is still
w orthwhile.
16
Table 1.2. D ielectric properties o f substrate materials for high Tc superconductor films.
f(GHz)
8'
tanS
T{°K)
Reference*
9.5
230
3 x l0 - 2
300
( 1)
3.2
300
3 x l0 -4
300
(2 )
300
310
3.2x10-2
300
(3)
1.3
1,800
lx lO -4
90
(2 )
300
1,900
5.8x10*2
80
(3)
33
38
4 x l0 - 3
300
(4)
10
28
4x10-3
300
(5)
300
26.1
1 .6
300
(3)
300
25.4
7.5x10-3
80
(3)
10
10
x l0 -5
300
(6 )
300
9.87
9 .1 x l0 -4
300
(3)
10
-
x 10 -6
77
(6 )
300
9.6
4 .2 x l0 - 5
80
(3)
10
11
300
(6 )
10
-
8.4x10 - 6
77
(6 )
9
-
1.5xlO - 8
77
(7)
72
-
4 .3 x l0 - 7
77
(7)
(0 )
500
9.3
1x10-3
300
(8 )
(e)
500
11.5
9x10*4
300
(8 )
L aG a03
0 .0 0 1
25
1.8x10-3
300
(9)
LaA 10 3
10
24
3.1x10-5
300
(6 )
10
15.3
5.8x10-4
300
(10)
10
-
7 .6 x l0 -6
77
(6 )
10
-
8.3x10*5
77
(10)
Substrate
S r T i0 3
Y SZ
M gO
A I2 O 3
continued on next page
1 .6
6 .2
2
x 1 0 -2
x l0 -5
17
T able 1.2 continued from the form er page
Substrate
f(G H z)
e'
tan5
T (°K )
R eference*
N dG a03
10
23
l.lx lO -3
300
(6)
10
-
3.2xl0*4
77
(6)
K T a03
25-37
2,000
2 x l0 -3
80
(11)
L iN b 0 3
1
39
4x10-2
300
(12)
M g F 2 (o)
300
5.45
6xl0*4
300
(8)
(e)
300
4.72
7 x l0 4
300
(8)
10
16
8.2x10-5
300
(6)
1.2 x l0 -5
77
(6)
< lx l0 4
300
(13)
lxlO*4
300
(14)
Y A 103
10
Si
-
-
12
10
3.78
S i0 2
o: O rdinary w ave,
e: E xtraordinary w ave.
* (1) T alvacchio et al., 1991;
(2) K obayashi e t al., 1989;
(3) G orshunov et al., 1988;
(4) T horp et al., 1973;
(5) L anagan et al., 1989;
(6) K o n a k a e ta l., 1991;
(7) B raginsky e t al., 1987;
(8) B ystrov e t al., 1986;
(9) S andstrom e t al., 1988;
(10) S im on e t al., 1988 & 1989;
(11) B elokopytov et al., 1990;
(12) X i e t al., 1983;
(13) S im on, 1989;
(14) von H ippel, 1954.
18
Table 1.3. Some basic properties of substrate materials for high Tc superconductor films.
Substrate
Crystal
structure
S r T i0 3
Lattice
constant
a-axis (nm)
Thermal
exp. coeff.
(ppm /°C)
Existence
o f twin
B uffer
layer
Melting
point
(°C)
Cubic (Perov)
0.3905
10.4
no
no
2,080
Y SZ
Cubic
0.5140
10
no
no
2,700
M gO
Cubic
0.4213
1 2 .6
no
no
2,800
A I2 O 3
Trigonal
0.4758
7
no
required
1,370
L aG a03
Ortho (Perov)
0.3888
10.3
yes
no
1,750
L aA 10 3
R hom (Perov)
0.3792
-1 0
yes
no
2 ,1 0 0
N dG a03
O rtho (Perov)
0.3837
8 .0 2
yes
no
1,580
M gF2
Cubic
0.4620
no
required
_
Y A IO 3
O rtho (Perov)
0.3662
_
no
no
1,900
Si
D iam ond
0.5431
4
no
required
1,150
S i0 2
Hexagonal
0.4998
0.55
no
required
-
19
1.3. Frequency D ependence o f M icrow ave Dielectric Properties
Since som e o f the m icrow ave m easurem ent techniques can m easure only one or
certain frequency points, great interest exists to use the dielectric properties/frequency
relation to estim ate the dielectric properties at frequencies other than the m easured
frequency points w ithout m aking additional m easurem ents. A ccording to the classic
dispersion theory, com plex dielectric perm ittivity e' and e" can be expressed in the
follow ing equations (W akino et al., 1986);
„
£ '(ra )
=
+
4 71 p : CO: ( CO: —
S
—
J
(C D j
s2 ^
1
-
CO
)
co
)
-------------+
(1.7a)
(YjCO )
2
4 tc p j C0j (Yj c o )
e (W) = ----2w— 2;------- 2-----------------------„2
2
2 -i2
J
(C0j
-
CO
)
+
(
1. 7 b )
(YjCO)
w here £« is the dielectric constant caused by the electronic polarization at higher
frequencies (&» = n2, n is the refractive index), coj is the resonant frequency o f dam ped
lattice oscillator (usually a>j > 1012), Yj and Pj are the w idth and intensity o f resonant peaks,
respectively. F o r co «
e’(co) =
e"(co) =
em
C0j, the equations (1.7) can be sim plified to follow ing equations:
+
X 4 7 i Pj
j
S 4 >t PjV i “
(1.8a)
(1.8b)
j
It can be seen that at co «
coj the dielectric constant is independent o f the frequency and the
loss tangent is proportional to the frequency w hich m eans fxQ = constant. It is still not clear
20
if the equation (1.8) is valid for all the m aterials at microwave frequencies. The validity of
equation (1.8) for som e o f the potential substrate materials will be exam ined in chapter 4.
1.4. Purpose o f Research
The final goal of this thesis is to obtain the dielectric constants and the loss tangents (or
quality factors) of potential materials for the substrate o f high T c superconductor thin films
applied at microwave frequencies. From the data on microwave dielectric properties, the
adequacy o f new substrate materials for m icrow ave applications can be judged.
T he correct choice o f microwave m easurem ent techniques is im portant as various
m icrow ave measurem ent techniques are available and various restrictions exist in the
techniques. For the substrate materials for H T S C thin films, low loss m easurem ent
capability o f the m icrow ave measurement m ethods will be m ainly required, w hile the
m easurem ent o f the dielectric constant is considered to be easier than the m easurem ent of
the low loss tangent. U nfortunately, although various low loss m easurem ent m ethods at
m icrow ave frequencies have been published, a systematic study on the m easurem ent
techniques for low loss m easurem ent is still not available. A system atic study and
com parison o f the low loss measurem ent techniques at m icrow ave frequencies is the other
goal o f this thesis.
T he existing low loss microwave m easurem ent techniques are still not perfect from all
points o f view as will be discussed in chapter 2. Great interest exists to improve the
existing techniques for low loss m easurem ent. The im provem ent o f a low loss
m easurem ent technique at microwave frequencies will be conducted in this thesis too.
T o exam ine the validity o f equation ( 1.8) on the potential substrate materials suggested
in this thesis is another goal o f this thesis.
21
The m icrow ave dielectric properties o f com posite material is a long-term interest but
the adequacy o f various pow der mixing rales at microwave frequencies is still a mystery.
A nother task o f this thesis is to com pare som e o f the m ost popular mixing rules at
m icrow ave frequencies. T he possibility o f using the com posite sam ple to estim ate the
dielectric properties o f pure material will also be investigated.
22
Chapter 2
M IC R O W A V E M E A SU R E M E N T TE C H N IQ U E S
The m icrow ave measurem ent techniques can be divided into tw o groups: i) Resonance
Techniques, and ii) Transm ission Techniques. The resonance techniques are the main
interest in this thesis and will be discussed in detail. A sim ple overview o f the transm ission
techniques will also be given to present an integrated introduction o f microwave
m easurem ent techniques. The resonance techniques do not have the swept frequency
capability. Only one or certain frequency points can be measured. All three space
dim ensions are considered for the com putation o f dielectric properties. Unlike the
resonance techniques, the transmission techniques usually have the swept frequency ability
for the measured frequency range. The transmission and/or reflection signals are always
tested to calculate the dielectric properties of the specimen. O ne dim ension (direction o f
signal propagation) is mainly considered. Usually, dielectric property measurem ent by the
resonance techniques has higher accuracy than measurem ent by the transm ission
techniques especially for the dielectric loss. That will be discussed later in this chapter.
A systematic com parison betw een available resonance techniques and a newly
im proved resonance m ethod will be given in this chapter. The resonance techniques can be
further divided into tw o categories: The first is that the resonance is basically supported by
the dielectric sam ple itself. The sample acts as a dielectric resonator. M etal shields with
different geometries are always introduced to prevent radiation loss. That type is called
dielectric resonance technique. It will be introduced in section 2.1. A m odification o f the
existing field distributions o f a dielectric resonator for dielectric properties m easurem ents is
given in section 2.2. Some parameters o f the dielectric resonance techniques will be
23
discussed in section 2.3. T he second type is that the resonance is supported by the m etal
w alls o f the m etal cavity. T he presence o f sam ples in the cavity causes only a
"perturbation" on the field distributions in the metal cavity. T he second type is called the
perturbation technique and w ill be given in section 2.4. Usually the dielectric resonance
techniques have higher accuracy on the m easurem ent o f the loss tangent than the
perturbation techniques. T he transm ission techniques w ill be introduced in section 2.5.
2.1. D ielectric R esonance Techniques
There are various dielectric resonance techniques for m easuring the dielectric
properties o f dielectric sam ples. The low est TE m ode o f a cylindrical dielectric sam ple is
alw ays used fo r m easurem ents because it is easy to identify the resonant peak, and the
calculation equations for the dielectric properties are more easily derived than those o f other
m odes. T h e m ain advantage o f the dielectric resonance m ethods is the higher accuracy o f
m easurem ents, but there are several disadvantages for the dielectric resonance techniques.
O nly a single frequency point can be m easured for each sam ple. T he resonance techniques
do not have the sw ept frequency capability. And, the calculations o f dielectric properties are
usually very com plicated; a com puter program is alw ays required to deal w ith the
com plicated B essel functions. The third disadvantage is the requirem ent o f sam ple
dim ension; m uch larger sam ple is needed than for the perturbation techniques. H ow ever,
the dielectric resonance techniques are still w idely used because o f their high accuracy o f
dielectric properties m easurem ents in com parison to other m ethods —especially for loss
m easurem ent. In this section, three different dielectric resonance m ethods w ill be
introduced: i) Post R esonance Technique, ii) C ylinder Cavity R esonance T echnique, and iii)
W aveguide R eflection R esonance Technique. T he differences betw een those techniques are
based on the different geom etrical arrangem ents o f m etal shields. T here is an other
24
dielectric resonance technique - "parallel plate resonance technique" which has special
advantages that the other techniques do not and will be given in section 2.2 where a
m odification to an existing model o f the field distributions will be given to improve this
method. All the measurem ents are mainly conducted by a HP8510A netw ork analyzer
system , as show n in Figure 2.1.
2.1.1. Post R esonance Technique
O riginally suggested by Hakki and Colem an in 1960, this m ethod has been widely
used and has becom e the m ost popular dielectric resonance m ethod for m easuring the
com plex perm ittivity of high dielectric constant and low loss materials in m icrow ave
regions. A cylindrical dielectric rod is placed betw een two parallel metal plates (highly
polished brass plates with diam eter 10 cm w ere used in m easurem ents o f this thesis) as
shown in Figure 2.2. Two coupling antennas were used to couple the pow er in and out.
The solid arrow s in the figure indicate the direction o f signal flow. The m ode chart is
drawn in Figure 2.3. The TEoi i mode was adopted for these m easurem ents because it is
easy to identify and isolate it from other modes and to compute dielectric properties from
m easured param eters. The TEoi i mode is one o f the interior modes; the resonant modes
are supported by the dielectric sample. M odes supported by the metal shield are called
exterior m odes and modes m ixed by the interior and exterior m odes are called mixed
modes. Since the metal shields o f the post resonance technique do not form a cavity,
basically, all the resonant modes in Figure 2.3 are interior modes. Figure 2.4 is a typical
resonant curve w ith transm ission signal (S 2 1) m easured. The resonant frequency fc, the
half pow er pow er bandw idth Af3 cjB, the insertion loss S21, and the dim ensions o f the
specimen w ere recorded for the calculation o f the com plex permittivity.
H P8510A N etw ork A nalyzer
Port 1
S ll
Port 2
S21
Device under test
Signal flow direction
Figure 2.1. M easurem ent setup.
26
Port 1
Port 2
Sample
M etal shield
(a) Side view
Signal flow direction
C oupling loop
(b) T op view
Figure 2.2. Post resonance technique.
27
S2 1
log
REF - 2 5 . 0 dB
4 . 0 dB/
MAG
REF 5
START
STOP
Figure 2.3.
1 0 .0 0 0 00 0 0 0 0
2 0 .0 0 0 0 0 0 0 0 0
GHz
GHz
Post resonator m ode chart o f a dielectric sample with dielectric constant 23,
diam eter 11.48 m m , and thickness 3.33mm,
28
0
S 21(dB )
S21
3dB
Frequency
Figure 2.4.
Typical resonant curve of the transm ission signal.
29
The dielectric constant is calculated by the following equations o f T E omn modes (M,
N = 0, 1 ,2 ....), (Hakki et al., 1960; Hennings et al., 1983; Kobayashi et al., 1 985a),
£'
=
(
— ) ( k ci + k co ) +
2 7t
1
(2.1a)
(2.1b)
Jo(kCj a>
J i ( k cia)
K o a K 0(k coa)
k c ia K ,( k coa)
(2.1c)
(2. Id)
w here kcj and kcQ are sample geom etry dependents, Jn(k d a) is the Bessel function o f the
first kind, Kn(kc0 a) is the m odified Bessel function o f the second kind, and a (=D/2) and L
are the radius and the thickness o f the specimen, respectively, as shown in Figure 2.2. The
N is for the T E omn mode. For any value o f kc0a, the M th solution k ^ a exists between uom
and U|m , w here J o( uom) - 0 and J i ( u im )= 0. As m entioned M, N=1 is adopted for
m easurem ent.
T he com putations o f dielectric loss and quality factor are given by,
B R, — C — R
Qu
Qu
(2.2a)
(2.2b)
(2.2c)
30
(2.2d)
J i ( k c ia)
K o(kc ia) K 2(k c ia) — K j( k c ia)
K ,( k c ia)
J i(k c ia) - J 0(k cja) J2(kc ia)
(2.2e)
( 2.20
w here S21 is the insertion loss, Q u is the m easured unloaded quality factor, Qd is the
quality factor due to the dielectric loss, R s is the surface resistance o f metal shields, and a is
the dc conductivity o f the m etal shields. The factor A is the ratio o f total energy stored in
the dielectric and air to the energy stored in the dielectric. U sually, the A factor is close to 1,
w hich m eans m ost o f the energy is stored in the dielectric. M ore detailed discussion on the
calculation o f factor A will be given later. The BRS, C, and R account for the conductor
loss, the loss o f the surrounding air, and the radiation loss, respectively. T he equation
(2.2a) can be explained as the total loss o f the system is the sum m ation o f the dielectric
loss o f the sam ple, the conductor loss o f the m etal shields, th e loss o f the surrounding air,
and the radiation loss. T he accurate calculation o f conductor loss is a critical point for
correct com putation o f dielectric loss. T he conductor loss w ill be discussed in the next
paragraph. T he loss o f the surrounding air depends on the am bient hum idity and the
density o f the sam ple. F o r a high density sam ple under low hum idity, the C can usually be
neglected. T he radiation loss can be alw ays neglected (H ennings et al., 1983; Kobayashi et
al., 1985a). A com puter program developed in the Intercollege M aterials R esearch
L aboratory (IM R L) o f T he Pennsylvania State U niversity (P enn State) w as used for the
calculations o f the dielectric constant and the dielectric loss.
31
For a specim en with quality factor higher than 2,000, the conductor loss is on the
sam e order as, o r higher than, the dielectric loss o f the specim en. T herefore, accurate
calculation o f the conductor loss o f the m etal shields is very im portant. U nfortunately, the
surface resistance R s o f the brass plates m ay change w ith the surface roughness, oxidation,
scratch, and tem perature variation. The m easurem ent o f the surface resistance o f m etal
shields is necessary. Tw o different m ethods o f m easuring the shield surface resistance
have been published. K obayashi et al. (1985a) suggested a m ethod using tw o cylindrical
sam ples cut from a sam e rod, one sam ple w ith thickness N tim es (N is for T E q in m ode)
the other sam ple. T he resonant frequency o f the TEoi i m ode o f the short sam ple is the
sam e as that o f the TE o in m ode o f the long sam ple. They derived the equation for
calculating the surface resistance from equation (2.2) based on the assum ption that the two
sam ples have the sam e loss tangent value at the sam e frequency,
2> g , 2 £ ' + w
N
, 1
1 ,
R s — 30 TC ( -) ------------------------ ( pr------ pr
)
Xo
1 + W N - l
Qui
QuN
( 2 31
{
w here Q u j and Q un are the m easured unloaded Q values o f the TEoi l and T E o in m odes
o f the shorter and longer sam ples respectively. The other m ethod w as suggested by the Xu
e t al. (1985). T hey derived the follow ing equation from equation (2.2) by using only a
single cylindrical sam ple w ith (D /L )2 « 3, w here the resonant frequencies o f TEoi 2 ar*d
T E 0 2 1 m odes are very close, so as the loss tangents,
A2
n
Rs =
Q u2
Bi -
Q ul
B.
(2.4)
32
w here A i and A 2 , B 1 and B 2 , and Q ui and Q u 2 are the factors and unloaded Q values in
equation (2.2) for T E oi 2 and TE 0 2 1 m odes respectively. The equation (2.4) is considered
to be an easier m ethod because only one sample is required for Rs m easurem ent. A fter the
surface resistance is obtained, the conductivity can be calculated from equation (2.20.
In addition to the m easurem ent o f dielectric properties, the extra m easurem ent on the
surface resistance is one defect o f the post resonance method. Dielectric loss low er than
5x1 O'4 is not recom m ended for m easurem ent by the resonant post technique unless the Rs
value can be precisely determined. T he other disadvantage o f the resonant post method is
the requirem ent for the sam ple's dim ension for this m ethod is largest am ong the four
dielectric resonance techniques. For instance, for a material w ith dielectric constant 20, a
diam eter 10 mm and thickness 5 m m specim en w ould be required for the m easurem ent at
10 G H z (The required dim ension will be sm aller for the other dielectric resonance methods
as w ill be discussed later). For a single crystal this dim ension can be quite difficult to
prepare. The Figure 2.5 gives the relationship am ong the sample's dim ension, dielectric
constant, and resonant frequency. In addition, m easurem ent can only be conducted on one
certain resonant frequency, while the cavity perturbation m ethod can m easure different
frequency ranges by using different w aveguides with different frequency bands (this will
also be discussed later). The main advantage for this post resonance m ethod is that
equations shown above for the calculations o f dielectric properties were well developed,
m uch sim pler than other dielectric resonance methods as will be m entioned later. In
addition, the accuracy o f those equations is high. Therefore, the technique is still used by a
lot o f people.
f D (GHz . cm)
100
8 = 10
£ ’=
20
8=
F= 40
0
2
4
6
8
D/L
Figure 2.5.
Relationship o f resonant frequency and sample dimensions o f the post
resonance technique.
10
34
2.1.2. C ylinder Cavity Resonance Technique
The setup o f the cylinder cavity dielectric resonance technique is show n in Figure 2.6.
The technique has been mainly used for Q factor m easurem ents on high Q materials
(Tam ura et al., 19B9; K onaka et al., 1991). The sample under test is put inside a copper
cavity with inner diam eter 3.0 cm and height 1.7 cm and on a styrofoam support, which
has a dielectric constant 1.03 and a loss tangent 1.5X10-4 at 10 GHz (Johnson, 1991, p.
166). Figure 2.7 is the mode chart. The T E oi5 m ode is used for m easurem ent. R esonant
peaks other than designated in Figure 2.7 are the exterior or m ixed m odes. The form ula
used for quality factor and loss tangent m easurem ent is,
&
a rf
-£
-
i
-
A
(2-5a)
(2Jb)
w here the definition o f factor A is the same as that o f the post resonance method.
Kobayashi et al. (1985) developed the equations for this system which can be used for the
calculations o f dielectric constant and dielectric loss. The equation is very com plicated and
needs a lot o f com puter work to solve it. The accuracy for the equations is still not clear.
For the calculation of loss tangent, if d/D and h/L are larger than 2, the conductor loss can
be neglected w ith loss tangent o f sample larger than 2X10-4 (Delaballe et al., 1981) to avoid
the com plicated com putation o f Qc. However, com plicated com putation o f dielectric
constant can not be avoided.
No radiation loss is one advantage o f the cylinder cavity m ethod as the specim en is put
inside a closed cavity. Since the electromagnetic fields decay fast outside the dielectric
(K obayashi et al., 1981), the conductor loss is m uch low er than that o f the post resonance
35
d=3.0cm
h=1.7cm
Port 2
Port 1
Specimen
Figure 2.6. C ylinder cavity resonance technique.
36
S21
REF
Iog
- 7 0 . 0 dB
1 0 .0 dB/
MAG
V
HEM 128
START
STOP
Figure 2.7.
5 .0 0 0 0 0 0 0 0 0
1 5 .0 0 0 0 0 0 0 0 0
GHz
GHz
M ode chart o f the cylinder cavity dielectric resonator w ith support thickness
6.1 mm and dielectric sam ple w ith dielectric constant 23, diam eter 11.48 mm,
and thickness 3.33 mm.
37
technique. The accuracy for Q m easurem ent should be higher because o f the lower
conductor loss. In addition, the sample dim ension requirem ents for the cylinder cavity
resonance method is sm aller than that for the post resonance method. For a sam ple with
dielectric constant 20, m easured at 10 G Hz, The required DxL is about 6 mmx3 mm —
m uch sm aller than the 10 mmx5 m m requirem ent o f the post resonance m ethod. Detailed
discussion on the sam ple dimensions will be given in section 2.2. The lack o f simple
equations to com pute the dielectric properties is one disadvantage. Furtherm ore, the
dielectric sample is put inside a closed metal cavity for the cylinder cavity method. The
TEoiS m ode o f the dielectric sample may be confused with o r interfered by the exterior and
m ixed modes which causing difficulty in identifying the desired resonant peak or
influencing the accuracy of m easurem ents. The higher the frequency o f T E 0 1 5 mode is, the
m ore serious o f the influence by exterior and m ixed m odes will be. A dielectric resonance
technique that com bines the advantages o f low conductor loss o f the cylinder cavity
m ethod, sim pler equations for field distributions, and w ithout the interference o f other
m odes for the post resonance m ethod w ill be given in section 2.2.
2.1.3. W aveguide Reflection Resonance Technique
In contrast to the above two dielectric resonance techniques where the transmission
signal is taken by the network analyzer, a reflection cavity dielectric resonance technique is
used in which case the reflection signal is measured. The setup is shown in Figure 2.8
w here only one o f the two ports o f the network analyzer is connected for test. The mode
chart is in Figure 2.9. T he T E ois mode was also used for this technique. The other higher
interior modes are confused with exterior or m ixed modes and are difficult to be identified.
Like the cylinder cavity technique, this m ethod has also been used for high Q factor
m easurem ents (O noda et ah, 1982; Plourde et ah, 1975). Again, the support is made by the
38
Port 1
tl
M oveable
shorted end
Specimen
1
Figure 2.8. W aveguide reflection resonance technique.
39
S 11
log
REF - 5 . 0 dB
1 .0 dB/
MAG
ftp
'/V \ M
A f
1 \ f v ^AA JA A,'JM
REF2
START
STOP
Figure 2.9.
6 .0 0 0 0 0 0 0 0 0
8 .0 0 0 0 0 0 0 0 0
A, A/
l ] 1W
11
!l
I
\ ,
GHz
GHz
M ode chart o f the waveguide reflection dielectric resonator w ith a C band
cavity a n d support thickness 7.6 mm. The dielectric sample has dielectric
constant 23, diam eter 11.48 mm, and thickness 3.33 mm.
40
styrofoam layer. T w o cavities are made by a C -band (3.95 to 5.85 G H z) and a X -band (8.2
to 12.4 GHz) copper w aveguides w ith one end shorted for m easurem ents on different
frequency regions. However, sam ples with resonant frequencies other than the tw o
frequency regions can still be m easured as long as the TEoig mode can be identified. The
position of the shorted end is adjustable for the best resolution o f the resonance signal.
Figure 2.10 show s the graph o f resonant trough. T he unloaded Q w as calculated by using
the expression (Sucher, 1963),
r\
_
t
—^min
P
P
*max~Px
^2
2
2 + VP
^ ^ m in
J* * P“ max
Af
A fx
the ± sign accounts for the undercoupled case (+) and overcoupled case ( - ) and Pmax~ 1T h e quality factor and loss tangent can be calculated by using the equation (2.5).
E xpression for the dielectric constant is not available for this method. T he conductor loss is
on the same order as that o f the cylinder cavity resonance technique. This reflection method
also has the advantages o f no radiation loss because o f the closed cavity and sm aller
required sample dim ensions. It has all the disadvantages that the cylinder cavity m ethod
has. In addition, m easurem ent error o f the reflection m ethod is higher than that o f the
cylinder cavity technique that will be discussed later.
2.2. Parallel Plate Resonance Technique
A m odification to a field model o f the parallel plate dielectric resonator will be given in
this section. A pplication o f this dielectric resonator for dielectric properties measurements
has the advantages o f low conductor loss and sm aller sample dim ensions as the cylinder
cavity and w aveguide reflection resonance techniques. It also has the capability of
41
Pmax= 1
Afx
Pm in
I
fo
Figure 2.10. Resonant curve of the w aveguide reflection technique.
42
calculating dielectric constant by simpler characteristic equation with no interference from
the exterior and mixed m odes as the post resonance technique.
2.2.1. Introduction
Different theoretical analyses on the parallel plate dielectric resonator in Figure 2.11
have been reported in several publications (Kobayashi et al., 1981; Kobayashi et al., 1985b;
G uillon et al., 1977; Jaw orski et al., 1979; Konishi et al., 1976; Itoh et al., 1977; Cohn,
1968; Yee, 1965; Pospieszalski, 1979). Like the cylinder cavity and w aveguide reflection
resonance techniques, the TEqis mode is the one always studied and the field expressions
are alw ays quite com plicated. An exam ple is the report by Kobayashi et al. (1981 &
1985b), w here the infinite series field expressions are used. T he accurate prediction o f
resonant frequency was given in their paper. They also calculated the conductor loss and
the radiation loss for d= » where the radiation loss can be neglected if the distance between
tw o metal plates is below cutoff (h< Aq/2;
is the free space w avelength corresponding to
the resonant frequency). In practical applications, the diam eter of metal plates is finite. The
radiation loss may becom e com parable w ith the conductor loss with h< X J 2 , especially
when the dielectric constant o f the dielectric is low, as will be discussed later. There are, to
my know ledge, two reports (Kobayashi et al., 1985a; Hennings, 1983) which deal the
radiation loss o f the post resonance technique (Lj = L 2 = 0), as mentioned before. Those
papers did not address the radiation loss for the case with finite diameter, and L [, L r £ 0.
The correct com putation for radiation loss w ith the finite diam eter 2b needs adequate field
expressions to account for the edge (r= b) o f m etal shields. Since the analysis requires quite
com plicated formulations, a simple model that can give accurate predictions o f the resonant
frequency, conductor loss, and radiation loss is greatly needed.
43
Lt
t
■H
T
z= L/2+L2
(5)
(3)
(5)
Er2
£r2
Er2
-
(6)
(4)
(6)
eo
£r'
(4)
(2 )
(4)
£rl
Erl
Erl
z= L/2
— z=0
8o
-----------------------z= -L /2
D
Figure 2.11. Parallel plate resonance technique.
z= - (L /2+L l)
44
Various sim plified models have been published (Itoh et al., 1977; C ohn, 1968; Yee,
1965; Pospieszatski, 1979). To confirm the accuracy o f the analyses, com putation of
resonant frequency was always included in the publications to compare calculations with
the experim ental results. The Itoh and Rudokas M odel (Itoh et al., 1977) is one o f the more
interesting ones. The accuracy for predicting resonant frequency by this model is not very
good. However, it has the great advantage o f using very sim ple field expressions. The
variational method suggested by the K ajfez (1986) to im prove the Itoh and R udokas Model
gives a much more accurate com putation o f the resonant frequency. Kajfez also computed
the conductor loss based on a perturbation method. U nfortunately, the Itoh and Rudokas
M odel is not adequate for the calculation o f radiation loss, as will be discussed later in this
section.
A goal of this thesis is to im prove the simple field expressions of the Itoh and
R udokas M odel to give the accurate prediction o f the resonant frequency and reasonable
com putations o f the conductor and radiation losses for the TEois mode w ithout using
com plicated field expressions. Im perfect field expressions are used in com putations. The
validity of the m odified model is established by com paring the results w ith the
experim ental values.
2.2.2. Field Expressions
Starting with the correctional results on the Cohn's M odel (Cohn, 1968) for TE ois
m ode (Kajfez, 1986), where fields in each regions are denoted by the subscripts I to 6:
E $i =
H 0 J , ( k cir ) c o s (P ,z - (J>i)
(2.7a)
c o s ( p ,^
e 02 =
E o ----------------------
sinh(a| Lj)
(2.7b)
45
c o s ( p , - +(J)2)
E i(>3 “
E 0 ------------------------- J i(kci r) sinh a 2(L 2 + - - z)
s in h (a 2
l
(2.7c)
J ,( k c ja )c o s (P 1y + (J > ,)
E if>4 =
E 0 ---------------------------------------- K 1(k cor ) s i n h a , ( z + L 1+ - )
K f(k coa) s in h (a j L j)
^
(2.7d)
E (J)5 =
J i( k c ia ) cos(p , ^ + <t>2)
r
E 0 ---------------------------------------- K I(k cor ) s i n h a 2 (L 2 - f - - - z )
K i(k coa) s in h ( a 2 L 2)
z
(2.7e)
E ° K j(k coa) K Kk c0 r)c o s (P iZ
(2 .7 0
4>,)
where
k Cj =
2.4048
a
k^■co
2 =
Pi“ - k o
(2.8a)
2
P, =
■2
k o^r
k0 =
« ^ o Eo
(f* i +
^2
^ 1.2 “
1
=
ta n
'1
(2.8b)
2
ci
(2.8d)
(2.8e)
0
,
,,
Q L
( — 1 ,-2 c o t. h a , 2 L
I 2) - P ( -
Pi
2
a i,2 ~
(2.8c)
2
k ci ~ k o e rl,r2
(2 .8 0
(2.8g)
and the J and K are the Bessel function and the m odified Bessel function, respectively. The
m agnetic field com ponents can be easily derived from the above equations by using:
VxH
=
j
CO
erE
(2.9)
46
Equation (2.7) is not a perfect field distribution. Expressions in regions 4 and 5 (Figure
2.11) do not satisfy the H elmholtz wave equation (Kajfez, 1986). The other concern about
those correctional results is the discontinuity at the boundary o f r= a because o f the
assum ption taken for the perfect magnetic conductor (PMC) in the Cohn Model. H ow ever,
after m aking som e modifications, it will be proved later in this section that the resonant
frequency and conductor & radiation losses can be reasonably calculated by using those
im perfect equations.
Instead o f using the PM C assumption, Itoh and Rudokas adopted a more realistic
situation by putting the continuity at the interface between region 1 and region 6 and, hence
the eigenvalue equation (Itoh et al., 1977),
—
J ,( k c ia)
“
"
^CQa K 0(k coa)
K ,(k coa)
<2 1 0 )
w here the equation (2.8a) is no longer valid. The equation (2.10) is exactly the sam e as the
equation (2.1c) of the post resonance method. The kci and kcQ are obtained by substituting
the equations (2.8b) and (2.8c) in equation (2.10). The fields in regions 4 and 5 are
postulated to be zero and the E ^i, E ^ , E ^ , and E^g are kept as those in the equation
(2.7). Therefore, discontinuities happen at the interfaces o f regions 4 and 5 with the other
regions.
In the w ork o f this thesis, in order to m ore accurately calculate the radiation and
conductor losses, all the Field expressions in equation (2.7) are adopted although defects
exist in the fields. The equation (2.10) is still valid for the continuities on the interfaces o f
p= a (betw een regions 1 &6, 2 &4, and 3 &5). However, it was found that the kcG value
obtained by equations (2.8b), (2.8c), and (2.10) was not adequate to obtain reasonable
radiation loss value because the decay of fields in regions 4, 5 and 6 will be too fast.
E xperim ental results reported in this thesis later will show that. The equation (2.8b) was
m odified to the following:
(2.11a)
(2.11b)
T h a t m odification is based on the fact that the distance betw een the two m etal shields
sh o u ld be sm aller than X J 2 to keep the system below cu to ff condition (K obayashi et al.,
1981 & 1985b; Jaw orski e t al., 1979). For h> X j l , the distance h is over the cu to ff
c o n d itio n and kc0 becom es im aginary, as can be seen from equation (2.11). T h e m odified
B essel function w ould be replaced by the H ankel function and the radiation loss
dram atically increased (K obayashi et al., 1985b). In order to keep the continuity at n=a, the
p is as show n in equations (2.7d), (2.7e), and (2.7f) w ould n o t be changed to P n, otherw ise
infinite series expressions w ould be needed in equation (2.7). A fter the m odification by
eq u atio n (2.11), except equations (2,8a) and (2.8b), all the equations from (2.7) to (2.10)
are still valid. The kcD obtained from equation (2.11) is then substituted into equation (2.10)
to com pute kci by a sim ple num erical m ethod. The resonance frequency is co m puted from
the equations (2.8e) and (2 .8 f) by a thickness m atching m ethod (Kajfez, 1986). The Q
factors due to the conductor and radiation losses are calculated by,
(2.12a)
(2.12b)
respectively, where the Pc, Pr, and W e are the pow er lost in conductor, the p o w e r lost by
radiation, and the total stored energy. T he W e and Pc are com puted by,
48
(2.13a)
i= 2
s
(2.13b)
For the calculation o f pow er dissipated by radiation, integration o f Poynting's vector over
the surface of r= b is made. More detailed discussion on Qc and Qr will be given in section
2.3.
2.2.3. Com putations and Experim ents
T o examine the validity of this simply m odified model, com putations and experim ents
were conducted for the parallel plate dielectric resonator. For all the experimental
confirm ations in other publications, dielectric constants larger than 35 were always
adopted. In our research, ceram ic specim ens with dielectric constants (£') 10, 23, and 37 are
used. The ceramic specim en with dielectric constant 23 has the quality factor due to
dielectric loss Qd = 81,900 / f(GHz), w hich has diam eter 11.48mm (D) and thickness
3.33m m (L). The other tw o with dielectric constants 10 and 37 have dim ensions DxL
12.66 m m x4.36 m m and 10.72 m m x3.06 m m , respectively. Tw o brass metal plates with
conductivity 1.41x10 7 S/m and diam eter 10 cm are used. The Qd and conductivity values
are the experim ental results which will be given in chapter 4. A styrofoam w ith dielectric
constant 1.03 (£rj) is used as the sample support and the e r 2 is equal to 1.0.
T he com putational and experimental resonant frequencies are shown in Figure 2.12.
The resonant frequencies o f the Itoh & Rudokas M odel and the variationally im proved Itoh
& R udokas M odel w ere com puted by the D R ESV 2 program (C opyright 1986, D.
Kajfez). The m odification in equation (2.11) gives dramatic im provem ent in com puting
18
•
Measurement
Modified method
Itoh & Rudokas Model
Variational improvement of I & R Model
16
f (G H z)
14
M=L = L
12
10
8
6
J
4
0
I
t
i
I
i
|____ |____ |____ |____ |____ |------ 1------ L
5
10
15
M (m m )
Figure 2.12.
Numerical and experimental results o f the resonant frequency o f the parallel
plate resonance technique.
50
resonant frequencies to the Itoh & Rudokas M odel. The accuracies are com parable to the
variational improvem ent m ethod given by the Kajfez. The accuracy o f this m odified model
is excellent at L j, L 2 < 5 m m and even better than the variationally im proved results. Such
results are surprising good for using such a sim ple model. Figure 2.13 is the com puted Qc
and Q r by using the the equation (2.12) o f the ceram ic specim en w ith dielectric constant 23
for h< X J 2 . By using the results in Figure 2.13, Figure 2.14 gives the calculated total
quality factor Q t and m easured quality factors o f the ceramic specim en, where
1
Qt =
1
1
1
Qd
Qc
Qi
(2-14)
From the experim ental results, at First the Q t values increase w ith increasing the distance
M betw een the shields and sample. That is expected because o f the decreasing conductor
loss. A fter the Q r values reached a peak, they decrease with increasing the M value
because the radiation loss starts dom inating the loss mechanism. It can also be found that
the com putational results by using the nonm odified Itoh & Rudokas M odel can not
correctly describe the radiation loss because the decay on the m odified Bessel function is
too fast. By m odifying the kc0 by equation (2.11), the decay o f m odified Bessel function is
closer to the practical situation, so as the resonant frequencies. The modification m ethod is
still not perfect; as the experimental results do not exactly fall on the theoretically calculated
solid curves in Figure 2.12 and Figure 2.14. The accuracy reported in the paper by
K obayashi et al. (1985b) is better than that o f this thesis. However, their field expressions
are considered m uch more com plicated than the present one. F o r such a simple model,
these results are considered to be quite satisfactory.
40
LOG( Qc & Qr )
35
30
Modified method
Itoh & Rudokas Model
lo g Q i
25
20
15
10
5
0
0
2
6
4
8
10
M(mm)
Figure 2.13.
Calculated Q , and Q,. by the Itoh and Rudokas Model and the modified method
of the parallel plate resonance technique.
Measurement
Calculated QT of modified method
Calculated QT of I & R Model
0
2
4
6
8
10
M(mm)
Figure 2.14.
Computed and experimental results o f quality factors o f the parallel plate
resonance technique.
53
2.2.4. M icrowave M easurem ents Using the Parallel Plate Dielectric Resonator
After solving the theoretical problem, the parallel plate dielectric resonance method can
be used to measure the dielectric properties. By m easuring the resonant frequency,
unloaded quality factor Q u, and all the dim ension parameters, the dielectric properties can
be calculated. The calculation o f the unknow n dielectric constant also uses the equations
(2.8e) and (2.80 by the thickness m atching method, like the computation o f theoretical
resonant frequencies in Figure 2.12 where the known dielectric constant was used to
calculate the unknow n resonant frequency. The quality factor Q and loss tangent can still be
calculated by the equation (2.5) with equation (2.5a) m odified by,
By com paring w ith other dielectric resonance methods, the main advantage o f the
parallel plate resonance method is that it has higher accuracy on measuring the quality
factor than the post resonance method and can also measure the dielectric constant,
although the accuracy may be lower for L j, L 2 > 5mm. T he conductor loss is about the
sam e order as that o f the cylinder cavity resonance method. T he equations for field
distributions for the parallel plate m ethod is much easier than that o f the cylinder cavity
m ethod. In addition, unlike the cylinder cavity and w aveguide reflection m ethods, the
dielectric sample is not put inside a closed metal cavity for the parallel plate technique. The
T E o is peak is easy to identify and has no interference from the peaks not supported by the
dielectric sample (exterior or mixed m odes) as shown in Figure 2.15. Furtherm ore, like the
cylinder cavity and the waveguide reflection methods, the requirem ent o f sample
dim ension for the parallel plate method in also less exacting than that o f the post resonance
m ethod. Figure 2.16 gives the relationship among the sam ple dim ensions and resonant
54
S2 1
l o g MAG
REF - 2 0 . 0 dB
4 .0 dB/
HEM
REF 5
START
STOP
5 .0 0 0 0 0 0 0 0 0
1 5 .00 0 0 0 0 0 0 0
GHz
GHz
Figure 2.15. M ode chart o f the parallel plate dielectric resonator with L j= L 2 = 4.6 m m and
dielectric sam ple w ith dielectric constant 23, diam eter 11.48 m m , and
thickness 3.33 m m .
20
D (GHz . cm )
16
8
12
=
10
e'= 20
8 —30
8
4
0
0
2
4
6
8
10
D/L
Figure 2.16.
Relationship of resonant frequency and sample dimensions o f the parallel
plate resonance technique with L ]= L 2 = 5 mm.
56
frequency for L j = L 2 = 5 mm, e ri= e r 2 = 1 by using the D RESV 2 program . The cylinder
cavity and waveguide reflection dielectric resonance methods have sample dim ensions very
close to that in Figure 2.16 if the side wall is far enough from the sample (d/D > 2 for the
cylinder cavity method). The only disadvantage o f the parallel plate resonance m ethod is
the high radiation loss can exist on low dielectric constant material, as will be discussed in
section 2.3.
2.3. O ther Param eters o f Dielectric Resonance Techniques
T he calculation o f factor "A" in equations (2.2), (2.5), and (2.15) will be given, the
conductor and radiation losses will be discussed, and the error analysis on the dielectric
constant and quality factor measurement w ill be introduced in the following.
2.3.1. Calculation o f the Factor "A"
From the definitions o f Q and Qd, the following equations can be derived,
tan 5
(2.16a)
(2.16b)
w = w d+ w a
(2.16c)
Pd =
( 2 .16d)
2 co W d tan5
where the Pd, Wd, and W a are the pow der dissipated in the dielectric, stored energy in the
dielectric, and the stored energy in the surrounding air, respectively. The factor A is then
expressed as:
w hich is the ratio o f total stored energy to the energy stored in the dielectric, as m entioned
before. From equation (2.13a), the stored energy is proportional to the dielectric constant
and the integration o f squared electric field. Since the electric field is exponential decay in
the air region, m ost o f the energy will be confined in the dielectric and A is very close to 1
if the dielectric constant is m uch larger than unit. There are, to my know ledge, two reported
m easurem ents that sim ply neglect the energy stored in the a ir region w ithout suffering
significant error on Q factor m easurem ents (Tam ura et al., 1989; Plourde et al., 1975).
B ased on a perturbation theory (H arrington, 1961), K obayashi et al. (1985b)
developed a new calculation m ethod for the factor A using the relationship betw een a
frequency shift (Af0) caused by a sm all change o f the dielectric constant (Ae1) o f a sample
w ith resonant frequency fQ and dielectric constant e'(
1 fo A e’
a
A
-
A f0 e’
(2.18)
T he advantage o f this equation is that it avoids the com plicated field com putations by using
the equation (2.13). The disadvantage o f the equation (2.18) is that a very accurate
calculation on the relation betw een dielectric constant and resonant frequency is required.
Figure 2.17 is an exam ple o f the com parison o f equations (2.13) and (2.18) for the parallel
plate resonance m ethod w ith 8 - 20, D= 12 m m , and L = 4 m m . G ood agreem ent is
reached betw een these tw o equations. Figures 2.18 and 2.19 give the relation o f factor A
w ith the diam eter/thickness ratio for the post resonance and parallel plate resonance
techniques respectively. For Figure 2.19, the L j= L 2 = 5 m m is adopted. T he exact sam ple
dim ension can be determ ined by referring to the Figures 2.5 and 2.16. For the cylinder
1.25
Eq. (2.13)
Eq. (2.18)
1.20
1.15
1.10
1.05
1.00
0
2
4
6
8
M (mm)
Figure 2.17.
Calculations o f the "A" factor by equations (2.13) and (2.18).
10
e’= 2
1
A = 1
0.9
10
D/L
Figure 2.18.
Relationship o f the factor "A" and the sample dimensions of the post
resonance technique.
V£>
f= 5 GHz
.6
f= 10 GHz
4
1.2
1.0
2
6
4
8
D/L
Figure 2.19.
Relationship of factor "A" and sample dimensions of the parallel plate
resonance technique with L j = L 2 = 5mm.
10
61
cavity and the w aveguide reflection resonance techniques, the A factors are the same order
as in Figure 2.19. In Figures 2.18 and 2.19, the A factor increases with decreasing the
dielectric constant because the transmission coefficient on the dielectric/air boundary is
larger and more energy is stored in the air region (larger W a/Wd ratio) for a low er dielectric
constant. The A factors of the post resonance m ethod are sm aller than those o f the parallel
plate resonance m ethod because o f the sm aller air region for the post resonance m ethod
w hich has a sm aller part o f the energy stored in the air region. For the parallel plate
m ethod, the curves w ith higher resonant frequencies have higher A values than those of
low er resonant frequencies because the sample dim ensions are sm aller for higher resonant
frequency. That causes the increase of the volume ratio of air region to the dielectric sample
and increases the A factor. It can be also found that the A factor decreases w ith increasing
the D /L ratio, which m eans the W a is mainly controlled by the side region (region 6) for
sm all D/L. For the post resonance technique, by using A = 1 for the calculation o f loss
tangent, the error is usually less than 5 % for dielectric constant larger than 10, as can be
seen from the Figure 2.18. However, by simply letting A= 1 for the other three dielectric
resonance techniques, the error will be higher than that o f the post resonance m ethod but
still can be controlled inside 10% if the dielectric constant and D /L are larger than 20 and 2,
respectively, w ith frequency significantly less than 10 GHz, as can be seen in Figure 2.19.
It has to be pointed out that the calculations o f factor "A" in Figures 2.18 and 2.19 are
based on the assum ption o f shield diam eter is close to ™. In real situation, the "A" factor of
the parallel plate resonance method should be m odified with changing the shield diam eter,
especially for low dielectric constant samples. For a sam ple with dielectric constant 5 at 10
G H z, the "A" factor may change from 1.3 to 1.6 for d= 3 cm to d= 10 cm .
62
2.3.2, Calculations o f Conductor and Radiation Losses
As mentioned, the Q factor due to conductor loss can be com puted by the equation
(2.12a). There is another method for the calculation o f conductor loss for the parallel plate
resonance technique. By following a perturbation method given by Kajfez (1984): If the
m etal shields are m oved a very small distance AM (M = Lj = L 2 ), the resonant frequency
w ill have a very sm all shift Af0, the quality factor from the conductor loss can be calculated
by (Kajfez, 1984; Kobayashi et ah, 1985b),
n
Q' =
f° ^
' A f. 8C
<Z-19>
w here Sc is the skin depth o f the metal shields. Like the equation (2.18), the advantage o f
this equation is that it does not need to deal w ith the field expressions as do the equations
(2.12a) and (2.13). T he disadvantage is the requirem ent o f accurate prediction o f the
frequency shift due to the distance change. Figure 2.20 is the com parison o f equations
( 2 .12a) and (2.19) o f the parallel plate resonance technique with sample thickness L= 4
m m . The larger difference between the results by using equations (2.12a) and (2.19) with
larger distance betw een the sample and shield agree with the results in Figure 2.12 where
m odified m ethod lost its accuracy for M > 5 mm.
T he relation o f Q c and Qr with the sample dimension, dielectric constant, and distance
betw een the shields o f the parallel plate resonance technique are shown in Figures 2.21 and
2.22 respectively. In Figure 2.22, the shield diam eter 10 cm was used. Figures 2.21 is only
a rough drawings and Qc values may have errors up to 15% for dielectric constants from 5
to 40. It can be easily understood that in Figure 2.21 the conductor loss increases with
increasing the D /L ratio because of the increasing surface areas o f the sam ple’s cross
section. W ith the sam e D/L value, the conductor loss increases with increasing the sample
Eq. (2.19)
Eq. (2.12a)
L= 4 mm
Qc
M= 5 mm
M= 1 mm
0
2
4
6
8
10
D/L
Figure 2.20.
Calculated Q . by using equations (2.12a) and (2.19).
On
L= 2 mm
2 mm
5 mm
- - D/L= 1
109
D/L= 2
D/L= 5
5 mm
10 mm
2 mm
10 mm
5 mm
105
10 mm
104
4
8
10
M (mm)
Figure 2.21.
Quality factor due to the conductor loss of the parallel plate resonance technique
with shield conductivity 1.41 x 10? IQ . -m.
-p*
Qr
8/4 \\ 8/2
16/4?
16/;
D /L =
16 mm/4 mm
16/8
10
M (mm)
Figure 2.22
Quality factor due to the radiation loss of the parallel plate resonance technique
with shield diameter 10 cm.
O'*
U\
thickness because the sample diam eter and the cross section are also increased. In Figure
2.22, the low er radiation loss for higher dielectric constant is because o f the low er
transm ission coefficient on the dielectric/air boundary and sm aller electric field amplitude
outside the sample. The higher radiation loss for sm aller D/L ratio can be explained by that
the larger side area for energy to be lost in sidewall. Some Qrs are not drawn to M = 10
mm because the c utoff condition is reached as m entioned in section 2.2. From Figure 2.21,
if the D /L< 3 and h / L > 3, the conductor loss can be neglected with loss tangent o f sample
larger than 2x1 O’4 which agrees w ith the results for the cylinder cavity resonance technique
given by D elaballe et al. (1981). The radiation loss can also be neglected under the
condition o f e ’> 10, L< 5 mm, and M < 6 mm, as can be seen in Figure 2.22. For e'< 10, in
order to keep low radiation loss, high conductor loss will com e up by reducing the
distances betw een shields and sam ple. In practical measurement, the choice o f sample
dim ension should ensure lower sum m ation o f radiation and conductor losses to have
higher m easurem ent accuracy.
2.3.3. Error A nalysis for D ielectric Constant M easurem ents
The m easurem ent errors o f dielectric constant m easured by the post resonance and the
parallel plate resonance techniques will be discussed in this section. There are three major
error sources for the dielectric constant m easurem ents. The first one (E el) is the error
caused by the frequency m easurem ent. The Eel can be easily derived from equation (2.8c)
which can be applied to both methods,
67
w here Af0 is the m easurem ent error on resonant frequency f0, not the frequency
bandw idth. This error for the H P8510A network analyzer is usually negligible.
The other error source is the m easurement error o f sample dim ensions. M easurem ent
errors on both diam eter D and thickness L can cause the error on dielectric constant
m easurem ent. The m easurem ent error (Ee2) from the diam eter m easurem ent for both the
post resonance and parallel plate methods can also be derived from the equation (2.8c),
2 k 2c i AD
Ee2 77
~
t r
<2 *2 1 >
o
The m easurem ent error (Ee3) from the thickness m easurem ent for the post resonance
can be expressed as (H ennings et al., 1983):
2
E e3 =
( 1 + W ) —- 2 —
4^
2 erL
(2'22a>
w here the param eters have the same meanings as defined in section 2.1.1. The Ee3 for the
parallel plate m ethod derived from equations (2.8e) and (2.8f) is,
e3
2 P i AL
_
L
fc-r o
(2.22b)
The Figure 2,23 gives the relationship o f Ee2 and Ee3 with the diam eter/thickness ratio
for the post resonance technique. The relationships o f the Ee2 and Ee3 and the sample
dim ension o f the parallel plate resonance technique are given in Figure 2.24. Again, the
exact sample dim ension can be determined from the Figures 2.5 and 2.16. W ith the typical
sam ple dim ension D /L= 2 to 4 for measurement, a 1% error on the d iam eter or thickness
E£2 /(AD/D) & Ee3/(AL/L)
2
1
0.5
0
10
D/L
Figure 2.23.
Normalized measurement errors of dielectric constant o f the post resonance
technique due to the error on dimension measurements.
Ov
CO
EE2/(AD/D) & E£3/(AL/L)
2
0.5
«•
I
L!
q
0
i
r
!
2
i
I
i
4
I
6
i
I
8
1----10
D/L
Figure 2.24.
Normalized measurement error o f dielectric constant of the parallel plate
resonance technique due to the error of dimension measurements.
70
m easurem ent can cause a m easurem ent error o f about 0.5 to 1.5% on Ee2 and E e 3 for both
the post resonance m ethod and the parallel plate method on the dielectric constant
m easurem ents.
The com bined rms error can be calculated by using,
(2.23)
w ith the assum ption that all errors are independent. Therefore, the total error on the
dielectric constant m easurem ent caused by a 1% error on the m easurem ents o f both
diam eter and thickness is about 1.5% for both the post resonance and parallel plate
resonance techniques. The definition o f E q w ill be given in the next section.
The last error source is the uncertainty enror. That error m ay arise for different reasons:
calibration error o f the network analyzer, temperature and hum idity instability, bends on the
m easuring cables, exact position and im perfection o f the specim en, and other unexpected
reasons.
2.3.4. E rror A nalysis for Q Factor M easurem ents
T he m easurem ent errors on the Q factor m easurem ents o f the dielectric resonance
techniques w ill be discussed in this section. There are various sources o f error for the Q
factor m easurem ents. The firs t error ( E q i ) is from the frequency bandw idth m easurem ent
error which can be expressed as,
B .) A( A f3dB)
E
q
,
-
( 1 +
(2.24)
Q d R s^")
A
AfjdB
7!
F or the H F8510A netw ork analyzer this bandw idth error A(Af3 (jBVAf3 dB is about 0.5%.
F or the cylinder cavity, the reflection, and the parallel plate resonance techniques, because
o f the low conductor loss, the m easured unloaded Q (Q u) is very close to Qd and BRS«
A/Qd- The E q i is m ainly controlled by the bandw idth error. However, for the post
resonance m ethod, the conductor loss term can not be neglected. Figure 2.25 gives the
relationship o f B /A and the sample dim ension for the post resonance technique. The higher
conductor loss term for higher dielectric constant exists because o f the sm aller sample
volum e and low er energy stored. For a material with dielectric constant 20, D/L= 2, and
Qd= 1,000 to 10,000, the E q i is about 0.6% to 2%, if the post resonance technique was
used to measure the quality factor at 10 GHz with shield conductivity 1 .4 1 x l0 7 / Q -m as
used for the m easurem ents in this thesis.
The second error (E q 2 ) is from the am plitude resolution error (AR in dB ). The error
can be calculated by using the expressions (Sucher, 1963),
E q2 =
(2.25a)
0.23 AR
w ith 3 dB bandw idth measured for the cylinder cavity and parallel plate resonance
techniques. How ever, for the post resonance technique, the above equation should be
m odified to,
E q2 =
(2.25b)
0.23 { I + Q d R s ® ) AR
accounting for its conductor loss. For the reflection m ethod, the amplitude resolution error
is (Sucher, 1963),
1 + Y
Y P max+ ^
(2.25c)
ma x
min
max
0.02
B /A
0.015
0.01
e '=
5
0.005
0
0
2
4
6
8
10
D/L
Figure 2,25.
Relationship of B/A and the sample dimensions of the post resonance technique.
IO
73
F or the cylinder cavity and parallel plate m ethods, the E q 2 is about 0.23% , where AA is
about 0.01 dB for HP8510A. From Figure 2.25, the E q 2 is about 0.3% to 1% fo r the post
resonance m ethod with the same conditions applied in the calculation of E q i and with Q j
also from 1,000 to 10,000. The amplitude resolution o f the reflection m ethod depends on
how sharp the trough is. E q 2 for reflection m ethod vary from 0.5% to 1.0% for the depth
o f the trough changing from 5 dB to 40 dB.
A nother error (E q 3 ) is from the calculation error on the conductor loss. For the
parallel plate, cylinder cavity, and reflection techniques, the conductor loss is pretty small,
this error is less than 0.5% . For the post resonance technique, the error on the conductor
loss m easurem ent plays a very important role. The E q 3 for the post resonance m ethod can
be expressed as (H ennings et al., 1983),
B AR,
E Q3 =
Q d R s^
( 2 .2 6 )
B y using the sam e dielectric constant and metal conductivity, a material w ith Q factor from
1,000 to 10,000 w ith 1% error on Rs m easurem ent, the E q 3 for the post resonance m ethod
is about 0.3% to 3%.
The m easurem ent error on the transm ission factor S21 is also one o f the error sources.
The E q 4 can be expressed as (Hennings et al., 1983),
A S21
E
q4
=
------------- ( 1
1-S21
B
+ QdR s7r)
A
( 2 .2 7 )
which indicates that a strong coupling betw een the device under test and the netw ork
analyzer will increase the error on Q factor measurement because the S 2 1 will approach 1
as the coupling increases. W ith a coupling w ith S21 less than -30 dB, the E q 4 is less than
74
0.5% . The total errors from the above four sources by using equation (2.23) are about 1.0
to 4% (for Qd from 1,000 to 10,000 with 1% error on Rs m easurem ent), 1.0%, 1.0 to
1.3% (w ith trough depth 5 to 40 dB), and 1% for post, cylinder cavity, reflection, and
parallel plate resonance methods, respectively.
A nother error is from the calculation error o f factor ” A” which is expressed sim ilar to
equation (2.24),
E q5 =
( 1 + Q dR s f - ) —
A
(2.28)
U sually this error can be neglected for high dielectric constant materials. However, for
dielectric constant less than 10, the role o f E qs can be important.
For the parallel plate resonance technique, the error from the calculation o f radiation
loss (E q 6) is an im portant factor which can be calculated by,
l Qd^Qi
AQr
(2.29)
Qr
A lthough the radiation loss also exists in the post resonance technique, it is considered
negligible, as discussed in section 2.1.1.
The last one (E q 7 ) is due to the uncertainty o f m easurem ents. As discussed in the
above section, that error is caused by different factors. The uncertainty o f measuring the
pow er level is im portant in the reflection m ethod w hich is calculated from (Sucher, 1963),
\l*2.
3/2
05
+ V)
7
(YPmax+ ^min)
^ mma xx “ ^ m i n
(2.30)
75
For Pmin «
Pmaxi the E q 7 is shown in Figure 2,26. It can be seen that, ideally, the Y value
should be larger than 0.2 to reduce the m easurem ent error, which m eans the difference
betw een Pmax afid Px is sm aller than 7 dB w ith Pmin«
Px- However, in a real situation,
the uncertainty error on the measurem ent o f Afx with Px close to Pmax can be much higher
than the error caused by the pow er level error. Therefore, it may be necessary to sacrifice
the low er pow er m easurem ent error to reduce the uncertainty error on Afx m easurem ent. A
1 % pow er level error can cause the uncertainty error E q 7 on the level o f 1.5 % for the
reflection m ethod for Y> 0,2. The loss from the styrofoam support can also cause
m easurem ent error and is considered to be negligible.
2.3.5. Liquid N itrogen Tem perature M easurem ents
The Q factors at liquid nitrogen temperature were measured by the cylinder cavity
resonance technique because the cavity is small and easily handled. The closed cavity can
exclude the influence o f vaporized nitrogen. The opened cavity was directly immersed
inside a container o f liquid nitrogen. Dry nitrogen gas was introduced into the container to
ensure negligible humidity around the specimen. The cavity was then covered and sealed
and liquid nitrogen was finally poured into the container to cool the cavity to liquid nitrogen
temperature.
2.3.6. Sum m ary o f D ielectric Resonance Techniques
The four dielectric resonance techniques have been system atically compared. The post
resonance method has well developed com putation form ulae for dielectric properties
m easurem ent. H ow ever its high conductor loss deteriorates the accuracy o f loss tangent
m easurem ent, and the required sample dim ension is the largest o f the four methods.
5.00
4.00
Cl,
<
>
3.00
a
2.00
1.00
0
0.2
0.4
0.6
0.8
1
Y
Figure 2.26.
Normalized measurement error o f quality factor due to the error on power level
measurement of the waveguide reflection resonance technique.
77
The cylinder cavity and reflection resonance techniques have good accuracy for Q
factor measurement; the field equations for the cavity resonance methods are com plicated;
com putation of dielectric constant by the waveguide reflection method is not available.
T he simple Itoh and Rudokas M odel for the parallel plate resonance m ethod has been
m odified in this thesis. The resonant frequency, conductor loss, and radiation loss have
been com puted to com pare with the experim ental results. A lthough self-consistent fields
are not constructed and the accuracy for this modified m ethod is not the best com pared
w ith other more com plicated field m odels, the presented m ethod gives a very sim ple way
to analyze the parallel plate dielectric resonator and still maintain acceptable accuracy in the
com putations o f resonant frequency, conductor loss, and radiation loss. The Q factor
m easurem ent is expected to be com parable to the cylinder cavity and w aveguide reflection
resonance techniques and better than the post resonance m ethod. Although the dielectric
constant m easurem ent is not better than the post resonance m ethod, the error is expected to
be acceptable, as seen from Figure 2.12.
Table 2.1 gives the comparisons o f the four dielectric resonance techniques for
dielectric constant and Q factor m easurem ents.
2.4. Perturbation Techniques
The fundamental concept of perturbation technique is that the presence o f a small piece
o f dielectric sample in the resonant cavity w ill cause a shift o f resonant frequency and a
decrease o f the quality factor of the cavity. It is also assum ed the change in the overall
geom etrical configurations o f the electrom agnetics upon the introduction o f sam ple m ust
be very small. The com plex perm ittivity o f specimen can then be calculated from the
changes o f resonant frequency and quality factor of the m etal cavity. The cavity can be
either rectangular or cylindrical as shown in Figures. 2.27 and 2.28. The sam ple is always
78
Table 2.1. Com parison between four dielectric resonance techniques.
Techniques
Post resonance
technique
C ylinder cavity
resonance technique
W aveguide reflection
resonance technique
Parallel plate
resonance technique
Advantages
D isadvantages
1. Good accuracy on dielectric
constant measurement;
2. W ell developed and sim ple
equations for field
distributions;
3. No interference o f exterior
and m ixed modes.
1. High conductor loss. Low er
accuracy o f dielectric loss
m easurem ent;
2. Accurate m easurem ent of
metal shield conductivity is
required;
3. Larger sam ple dim ension
required.
1. Low conductor loss. H igh
accuracy on dielectric loss
m easurem ent;
2. Sm aller sam ple dim ension
required;
3. N o radiation loss.
1. C om plicated expressions for
field distributions;
2. Interference o f exterior and
mixed m odes.
1. Low conductor loss. H igh
accuracy on dielectric loss
m easurem ent;
2. Sm aller sam ple dim ension
required;
3. N o radiation loss.
1. M easurem ent o f dielectric
constant is not available;
2. Interference o f exterior and
m ixed m odes.
1. Simple equations for field
distributions;
2. Low conductor loss. H igh
accuracy on dielectric loss
m easurem ent;
3. Sm aller sam ple dim ension
required;
4. No interference of exterior
and m ixed m odes.
1. Existence o f radiation loss.
79
a
Port 2
Sam ple
Port I
Coupling hole
Figure 2.27. Cavity perturbation technique using rectangular cavity.
80
Coupling
loop
Sample
Port 1
Port 2
a
1<
. -
H
Figure 2.28. Cavity perturbation technique using cylindrical cavity.
81
put in the position o f m axim um electric field for testing, w here the equations for the
calculations o f dielectric constant and loss can be derived more easily than by putting the
specim en in the non-m axim um position.
2.4.1. Rectangular Cavity
F o r the rectangular cavity, the T E ion m odes (N is integer) are widely used for the
com plex perm ittivity m easurem ents. A sm all piece o f rod, sheet, o r bar shaped sam ple is
placed in the position o f maxim um electric field. Usually there are N positions in the cavity
w ith m axim um electric field for the T E jon modes. Odd m odes (N: odd) are m ore often
used because the geom etrical center is alw ays one o f the m axim um electric field positions
despite the fact that the m axim um field positions change w ith changing N values. For even
m odes, the m axim um electric field positions are more difficult to locate because the center
position is no longer one o f the positions o f m axim um electric field. The specim en position
w ill have to be m oved w ith different T E ion modes.
T he com plex perm ittivity is calculated from the changes o f resonant frequency and
quality factor. The theoretical calculations o f resonant frequency fc and quality factor Q for
T E ion m odes o f the rectangular cavity are by the following two equations,
N .2,1/2
(2.31a)
( k a d ) 3 bq
Q =
(2.31b)
w here c is the speed o f the light in free space, T| is the intrinsic im pedance, Rs is the surface
resistance o f the cavity, and a, b, and d are the dimensions as show n in Figure 2.27. The
above two equations can be found in a lot o f electrom agnetic wave literature. The m easured
Q value is calculated by the following equation:
Q =
f°
Af
11-S21I
A r 3dB
(2 -3 2 )
where Af3 dB is the half pow er bandw idth o f the resonant peak at fc.
The basic equation o f cavity perturbation theory was derived by A ltschuler et al.
(1963),
/
fc- f s
fs
1
7
J VQc
1
Qs '
E , E 2dV
' vs
V~ r
f
,2
JFvv r
E . I dV
(2 .3 3 )
c
where fc and fs are the resonant frequencies, and Qc and Qs are the quality factors o f the
cavity w ithout and with sam ple inside the cavity, respectively; Vc and Vs are the volum es
of cavity and the sample, and E i and E 2 are the electric fields in the cavity and sample,
respectively. From equation (2.33), the changes o f fc and Q can be expressed as,
83
As m entioned before, the basic assum ption for the cavity perturbation technique is that
the change o f electrom agnetic field upon the introduction o f the sample must be small.
From the assum ption, by inserting the sample sym m etrically in a region o f m axim um
electric field, the com plex permittivity is calculated by (Dube et al., 1988),
V c < fc - f s )
£
=
'
2 V ,f .-
+
‘
( 2 ' 3 5 a)
(2 -3 5 b )
The dielectric loss tan8 and quality factor Q due to the dielectric loss o f the m easured
m aterial can be expressed as:
1
—
Q
=
tan5-
=
e"
—
(2.36)
The device under test is connected to the H P8510A network analyzer for testing the
transm ission signal as shown in Figure 2.1. Tw o copper cavities were fabricated by
standard W R -90 and W R-42 w aveguides for m easurem ents on X band frequencies (8.2 to
12.4 GHz) and K band frequencies (18 to 26.5 GHz) respectively. The lengths o f the X
band and K band cavities are 13.5 cm and 10.4 cm respectively. Tw o copper plates are
placed on both sides o f the each waveguide to form a resonant cavity. A circular hole or
longitudinal slot w as drilled o r cut on the center o f each copper plate for coupling the signal
in and out o f the X band cavity. A typical resonant mode chart on the screen o f netw ork
analyzer for the X band cavity is show n in Figure 2.29.
Since looking for high Q m aterial is one o f the m ajor objectives, to increase the
sensitivity for Q factor m easurem ent, the Q value o f the cavity should be as close to the
84
S21
1o g MAG
REF “ 7 0 . 0 dB
1 0 . 0 dB/
hp
TI HOs
!|
A
v
1
J
A
J V ■ In / r W
Iff'T r
TE 05
TE
TEioi
|
TEio i
1
refO
j
K . ii
l l 'l "
IT
START
STOP
8 .2 0 0 0 0 0 0 0 0
1 2 .4 0 0 0 0 0 0 0 0
GHz
GHz
Figure 2.29. Mode chart of the X band rectangular cavity with length 13.5 cm.
85
ideal (theoretical) Q value in equation (2.31b) as possible. The choice o f the dimensions o f
the hole or slot for coupling the signals in and out o f the cavity is im portant and will
influence the cavity quality factor dramatically. Different coupling hole diam eters were
used to com pare the measured Q values o f the cavity. Com parisons o f the theoretical Q
values and the practically measured Q values with various coupling hole diam eters for the
X band cavity are listed in part (a) of Table 2.2. For larger diameters, the too much energy
was lost from the coupling holes to get good Q values. F o r sm aller diam eters, the
measured Q factor o f the w eak signal is deteriorated by the noise level. It is found that hole
diam eter 2.9 m m gives the best Q values for the empty cavity, which are about one half o f
the theoretical Q values obtained from the equation (2.3 lb ). This coupling hole diam eter
w as then used for the m easurem ents at X band frequencies. About the K band cavity, a
coupling slot w ith width 1.0 m m , w hich gives measured Q factors a little less than one half
o f the theoretical Q values as show n in the part (b) of Table 2.2 was used. Copper
conductivity 5 .8 x l0 7/Q -m was used for the calculating the theoretical Q values in Table
2 .2 .
2.4.2. Cylindrical Cavity
For the cylindrical cavities, T M 010 mode of the cylindrical cavity is usually used. The
sam ple was inserted along its sym m etric axis in the position o f m axim um electric field
strength for easy measurem ent and calculation as show n in Figure 2.28. The theoretical
resonant frequency and quality factor o f the TM oio m ode o f the cavity can be calculated as
(Ramo, et al., 1984, p. 495),
fc =
c
11-5
a (cm)
GHz
(2.37a)
86
T able 2.2. C om parison o f m easured q uality factors (Qm) and theoretical q uality factors
(Qth) o f two rectangular cavities used for the cavity perturbation technique (TE
m odes).
(a) X band cavity (length= 13.5 cm )
8.59 G H z
(T E 1 0 5 )
10.6 G H z
(T E 1 0 7 )
11.9 G H z
(T E 1 0 9 )
t= 2.2 m m
3,270
2,530
3,060
2.5 m m
4,350
4,070
4,560
2 .9 m m
4,630
4,590
5,470
3.3 m m
3,360
2,540
1,250
5.0 m m
1,820
1,400
494
Qth=
9,020
10,640
12,240
Qm
t: D iam eter o f the coupling hole.
(b) K b a n d cavity (length= 10.4 cm , coupling slot w idth= 1.0 m m )
1 9 .1 G H z
2 1 .1 G H z
2 3 .4 G H z
2 5 .8 G H z
(T E 1 0 9 )__________ ( T E i o n ) ________ ( T E i o n ) _______ ( T E 1 0 1 5 )
Qm
2,550
2,570
3 ,330
3,480
Qth_____________ 6,120___________ 6,710____________ 7 ,4 4 0___________8,070
87
(2.37b)
where a and d are the inside radius and height o f the cavity, respectively. The resonant
frequency depends only on the radius o f the cavity because the field varies only in the r
direction. T he equation (2.33) used by the rectangular cavity can also be sim plified to
calculate the dielectric constant and the loss value for the cylindrical cavity (N akam ura,
1960),
e' =
0 .5 3 9
Vc( f c- f s )
c
— +
v sfs
e
(2.38a)
(2 .3 8 b )
O ne disadvantage o f the cylindrical cavity m ethod is that only a single frequency point can
be m easured for one cavity, w hile the rectangular cavity has several T E jon m odes fo r
m easurem ents.
T he precision o f the cavity perturbation technique is very difficult to control, especially
for the m easurem ent o f loss tangent. C hao (1985) has discussed the uncertainty o f the
cavity perturbation technique by using a rectangular cavity w ith cylindrical sam ples. H e
found that the theoretical errors due to the m easurem ent error on resonant frequency and
quality factor in com plex perm ittivity m easurem ent decrease w ith increasing the radii o f the
sam ples. H ow ever, under his analysis, the electric field w as assum ed uniform in the
sam ple, w hich is not correct w hen the sam ples becom e larger. It w as reported from the
other source (A S T M , 1986) that a cylindrical specim en w ith d iam eter 1.04 m m u sed in the
X band frequencies gave the m ost accurate m easurem ent results.
88
M easurem ent errors can also be created by other sources. For dielectric constant
m easurem ent, from equations (2.35a) and (2.38a), the error is m ainly from the
m easurem ent errors on frequency shift and volum e ratio. W hen carefully handled, the error
from frequency shift m easurem ent can be controlled to be less than 0.5%. The error from
volum e ratio m easurem ent is considered to be m ore difficult to control because o f the
difficulty o f precise m easurem ent on the dim ensions o f sm all specimens. T hat error
depends upon the dim ensions and shape o f the specim en. For the loss m easurem ent, from
equations (2.35b), (2.36), and (2.38b), the loss tangent is basically independent o f the
volum e ratio if dielectric constant is much larger than 1. The error o f loss tangent
m easurem ent is dom inated by the frequency shift error and especially the quality factor
change error. F o r the consideration o f m easurem ent lim itation on dielectric loss, by using
cavity with quality factor 5,000 at empty cavity, ideally the H P8510A system can detect the
change o f the Q value o f the cavity by inserting a rod sam ple w ith dielectric loss lx lO -4
and diam eter 1.0 mm. H ow ever, because the com bination o f different factors —
hom ogeneity o f the specim en, coupling holes, specim en holes, and the uncertainties o f the
m easurem ent setup and the netw ork analyzer — the accurate m easurem ent on Q value was
lim ited to about Q< 1,000-2,000. As much as 50% dielectric loss m easurem ent error was
reported on an alum ina sample (tanfi- 4 x 1 0 ^ ) m easured by the perturbation technique
(Dube et al., 1987) using the sam e X band rectangular cavity dim ensions m entioned in this
thesis. For best measured results, the surface o f the sam ple should be well polished.
H um idity is another factor known to influence the measured accuracy. Dry nitrogen gas
was introduced to keep the test under a low hum idity environm ent. The lim itation on
dielectric constant m easurem ent is about e'< 100.
A lthough the cavity perturbation technique has the limitation of low accuracy on high
Q m easurem ents, the technique has several additional advantages:
89
1. Unlike the dielectric resonant techniques measurements, there is really no severe
tolerance lim it on the shape and dimension o f measured specimen;
2. R equirem ent for specimen size is very small, therefore the sam ple is easily prepared.
Som e m icrow ave techniques require large sam ple (for exam ple, post resonance
technique) which may be quite difficult to prepare, especially for large single crystals.
In addition, due to the thin rod shape o f fibers, perturbation technique is the m ost
suitable m ethod for the measurements o f dielectric properties o f fiber samples at
m icrow ave frequencies;
3.
Unlike the dielectric resonance techniques where com puter program s are required to
solved the com plicated characteristic equations, the calculations for the complex
perm ittivity o f the perturbation technique is relatively simple and does not require the
com puter program .
Since the perturbation m ethod w ith the cylindrical cavity has the disadvantage that only
a single frequency point can be m easured while the rectangular cavity can m easure several
points, the rectangular cavity was adopted instead o f the cylindrical cavity for our
m easurem ents on thin rod o r b ar shaped samples.
2.4.3. Re-entrant Cavity
A different setup o f perturbation technique is shown in Figure 2.30. An electroded
sam ple is placed at the end o f a cylindrical cavity. The method is called the re-entrant cavity
technique. The electric field in the sam ple is assum ed to be uniform. For high dielectric
constant o r high frequency conditions, that assumption will no longer be valid because o f
the small w avelength under those tw o conditions. Calculations o f dielectric properties are
Sam ple
M etal
sam ple
holder
Coupling
loop
Port 1
Port 2
Figure 2.30. C avity perturbation technique using re-entrant cavity.
91
based on the circuit theory. The resonance occurs while the im pedance o f the sam ple is the
sam e as the impedance o f the cavity with opposite sign,
j Z 0 tan fl/
=
-l/jQ )C g
(2.39)
w here Z q is the characteristic impedance o f the cavity, I is the cavity length, and Cg is the
capacitance o f the gap region as a parallel plate capacitor. In the gap region where the
sam ple is placed, the capacitance values changed with sample put in or not at the end are
expressed as (Lanagan, 1987),
C gs = Cs + C as + Cf
(2.40a)
C go = Ca + Cf
(2.40b)
relatively, where Cgs and Cg0 are the capacitances in the gap region with and w ithout the
sam ple, respectively, and Cas, Ca, and Cf are the capacitances o f the annular area around the
sam ple, o f the air gap, and o f the fringe, respectively. The dielectric constant can be
calculated from the equations (2.39) and (2.40). T he loss tangent is calculated by,
,a n S
= k
=
£
■ i
(Z4I)
w here Q s and Qc are the quality factors o f the cavity with and w ithout the sam ples at the
sam e frequency, respectively. For the m easurem ent o f Qc, the center conductor o f the
cavity w ithout the sam ple m ust be adjusted to the same frequency as that o f the cavity with
the sam ple. In the IM R L o f Penn State, tw o cavities were built with resonant frequencies 3
G H z and 5.6 GHz and quality factors around 500. The lim itations on m easurem ents o f the
dielectric constant and dielectric loss are e '< 25 and tan5> 2x10 '3. The m ethod is not
adequate for our m easurem ents because o f its m easurem ent limitations.
92
2.5. Transm ission Techniques
The main advantage o f the transmission techniques is the swept frequency capability.
Like the resonant techniques, the transm ission techniques can be further divided into two
categories too. The first is the distributed transm ission method, in which the S 11 and S 2 1
param eters are m easured for the calculations o f dielectric properties. The other is the
lum ped impedance m ethod in which, like the re-entrant method, the calculations o f
dielectric properties are based on the circuit theory and the electric field in the measured
sam ple is assum ed to be uniform .
2.5.1. D istributed Transm ission Technique
A transverse loaded dielectric slab or disk is usually bounded by a w aveguide for the
distributed transm ission m ethod, as shown in Figure 2.31. The S I 1 and S21 param eters
can be expressed as (Ligthart, 1983):
S ll
(2.42a)
2
1 - p e
—2 YI
(2.42b)
Y
(2.42c)
(2.42d)
93
W aveguide wall
S21
Port 1
Sample
Figure 2.31. Distributed transmission technique.
94
(2.42e)
k 0 = coX pe^
(2 .4 2 0
w here y is the propagation constant in the sam ple, p is the reflection coefficient on the
boundary, kc is the w aveguide cu to ff w avenum ber, and p r is the m aterial's relative
perm eability. The equations (2.42) can be further sim plified to the follow ing fo r the
calculations o f dielectric constant and loss factor:
(X )2 Yo
( 1 - S l l ) 2 - S 2 12
( 1 + SI 1 )2 - S 212
(2.43)
T he application o f equations (2.42) and (2.43) to calculate the dielectric properties needs
both S I 1 and S21 to be m easured. T his technique is called " S l l & S21 transm ission and
reflection technique" or "SI 1 and S21 technique" for sim plification. T he lim itation on the
m easurem ent o f dielectric constant is based on the thickness o f the sam ple. T he thickness
should be less than one h a lf o f the w avelength to avoid resonance. T he u p p er lim it m ay be
about 1,000. H ow ever, the loss m easurem ent is only valid fo r tanS> 10~2. F o r m aterial
w ith loss tangent less than 10'2, the decay for the S l l and S21 is too sm all to be precisely
detected by the netw ork analyzer.
T he experim ental setup for the S I 1 and S21 technique is also connected to the
H P 8510A netw ork analyzer, as show n in Figure 2.1, and the S I 1 and S21 data are
m easured w ith frequency from 8.2 G H z to 12.4 G H z controlled b y a H P9816 C om puter
(not show n in Figure 2.1). T he sam ple holder is m ade by a sm all section o f X band
w aveguide w ith thickness larger than or equal to and the sam e order as the thickness o f the
specim en. T he sam ple is cut w ith the sam e dim ension as the w aveguide inside cross
section (0.9 in x 0 .4 in) then the sam ple is slid into the sam ple holder, as show n in Figure
95
2.32. The dielectric constant and dielectric loss are calculated from equations (2.42) and
(2.43). A com puter program is used for the calculation. The m easurem ent accuracy
depends upon the precise measurements o f S I 1 and S21 data, therefore calibration m ust be
conducted before the test.
Lanagan et al. (1988) simplified the equation (2.42b) for high dielectric constant (e'>
100) material,
S21 = ( 1 —p 2 )e~
= IS 21lej9
(2.44)
w here the log S21 is linearly related to the frequency. The dielectric constant and loss can
be calculated from the slopes o f phase and amplitude o f S21 (Lanagan et al., 1988),
e
=
. A0
c 2
<T 7 TTT)
A f 2 Tt t
_
tano =
(2.45a)
Al S211 8.686 c
---------------------------------------------------------------------------------------- (2.45b)
The m ethod is only suitable for high dielectric constant m easurem ent and the loss tangent
m easurem ent is lim ited to tan8> 0.1. The m ain advantage for using equations (2.45) is that
only S 2 1 data are needed for dielectric properties calculations. By using only the S 2 1 data,
the calibration procedure is much sim pler than that o f calibrating both S 11 and S21 signals.
There is another m ethod of using only the S21 data for calculating the dielectric
constant (Lanagan, 1987):
2
£'
”
<7 " >
A-d
^.n
+ ( 7“ >
c
2
(2.46)
96
Spiig
Sam ple
W aveguide
Figure 2.32. Setup o f device under test o f distributed transmission technique.
97
where Xo is free space wavelength at frequencies with m axim um S21 values, Xd is the
dielectric w avelength, and Xc is the w aveguide cutoff wavelength. The sam ple thickness is
equal to nXd/2 where n is an integer. The dielectric constant is calculated by equations
(2.42b) and (2.46). The iterate method is used to find dielectric loss values w hich meet the
m easured S21 data. One disadvantage for using the equation (2.46) is only som e certain
frequency points can be measured. The advantage o f swept frequency capability on
distributed transm ission techniques is sacrificed.
Dube et al. (1987) suggested a different setup for the measurem ent o f dielectric
constant and loss factor as shown in Figure 2.33. They derived the following characteristic
equation from an inhom ogeneous filled rectangular w aveguide (Collin, 1990, p436) with
very thin sample thickness,
2
2
2
1
2
2
tan[ d ( k 0 - y ) ] tan[ - c ( e r k 0 - y ) 3 =
2
r
2 >'2
(k0- y
2
( e r k0 - y
)
21/ 2
)
(2.47)
T he calculation form ulae for the dielectric constant and loss can be derived from that
equation. The m easurem ents of dielectric properties are lim ited to e'< 10 and tan5> 10*3.
2.5.2. Lum ped Im pedance Technique
A sm all disk-shaped and electroded sample is placed on the end o f a shorted coaxial
line, as show n in Figure 2.34. Complex reflection coefficients are measured by the
im pedance analyzer or network analyzer. The dielectric constant and loss can be obtained
by the form ula o f com plex reflection coefficients (Stuchly, 1974),
98
/
Port 2
>H h<
Port 1
Sam ple
W aveguide
Figure 2.33. Dube transmission technique.
99
C enter conductor
‘Specim en
O uter conductor
Figure 2.34. Lumped impedance technique.
100
r* = re
-j9 _
1
1 + jcflCoZ 0 £r
(2.48)
where CG= \ z j l is the free space capacitance, A is the sample area, t is the sample
thickness, Zq is the characteristic impedance o f the coaxial line, er is the complex relative
perm ittivity, £q is the free space permittivity value, and the T * is the com plex reflection
coefficient with amplitude T and phase 9.
The lum ped im pedance m ethod has also assum ed the uniform field in the sample.
Therefore, measurements o f high dielectric constant material or at high frequencies are not
adequate. The lumped im pedance method is reliable only for frequency and dielectric
constant up to 2 GHz and 100, respectively. The loss tangent can m easure down to about
10- 2 .
For application as substrate materials o f HTSC thin films, low loss is always required.
B ecause o f the limitation o f low loss measurem ents by the transm ission techniques, they
are not adequate for m easuring the loss tangent o f substrate m aterials in this research.
However, transm ission m ethods are still valid for m easuring dielectric constant. It w ill be
interesting to com pare the m easured results o f dielectric constant by transmission
techniques and resonant techniques. In this thesis, the S I 1 and S21 m ethod will be used in
chapter 5 to m easure the dielectric constants o f some com posite sam ples for confirm ing
and com paring with the results obtained from the resonant techniques.
2.6. C onclusion o f M icrow ave M easurem ent Techniques
A general review o f m icrow ave m easurem ent techniques has been given in this
chapter. T able 2.3 gives the sum m ary o f the different techniques discussed in this chapter.
D ifferent techniques were designed for the m easurem ents o f different dielectric constants,
101
Table 2.3. Sum m ary o f m icrow ave dielectric m easurem ent techniques.
M ethod
Post
Dielectric
constant
limitation
Dielectric
loss
limitation
Frequency
limitation
Sam ple
geom etry
Swept
frequency
capability
broad range
>1x10*4
<26 G H z
cylinders no
no
resonance
C ylinder cavity
electrode
broad range
> lx lO -5
<26 G H z
resonance
W aveguide
broad range
> l x l 0 '5
<26 G H z
broad range
> 1 x 1 0 -5
<26 G H z
<100
> 5 x 1 0 -4
<26 G H z
<100
> 2 x 1 0 -3
<25
> 2x10*3
thin rod or
GHz
bar
at 2.5 GHz
electrode
broad range
> 1 0 -2
<26 G H z
broad range
>10'
<26 G H z
no
no
rectangular
yes
rectangular
yes
for X band
<10
>10*3
<26 G H z
rectangular
yes
thin slice
transm ission
Lum ped im pedance
yes/no
for X band
transm ission
Longitudinal
no
small disk
and transm ission
S21
thin rod, bar,
at 3 & 6
cavity
S 11 & S 2 1 reflection
cylinders no
or slice
perturbation
Re-entrant
no
electrode
perturbation
TM cavity
cylinders no
electrode
resonance
T E cavity
no
electrode
reflection resonance
Parallel plate
cylinders no
<100
> 1 0 -2
<2 G H z
elec traded
small disk
yes
102
loss tangents, and sample geometries. Lim itations for various techniques should be
considered to choose the most suitable techniques for different purposes. Since low loss is
one o f the m ajor requirem ents for the high Tc substrate materials, the four dielectric
resonance techniques that can m easure lower loss tangent than other techniques w ere
chosen for the purpose o f low loss m easurem ent. Because som e o f the single crystals were
grow n to the shape o f Fibers, the perturbation techniques are the m ost suitable fo r the thin
rod shaped fibers. By com paring TM cavity and TE cavity techniques, the TE cavity was
chosen because it can m easure more frequency points than the T M cavity technique. In
addition, one transm ission method, S l l and S21 transmission and reflection technique,
was chosen ju st to confirm the dielectric constant data measured by the other resonant
methods. The experim ental results will be given in the chapters 4 and 5.
103
C h a p te r 3
SE L E C T IO N A N D PR EPA R A TIO N O F N EW SU B ST R A T E M A T ER IA L S
The search for new substrate materials for the applications o f high Tc superconductor
thin films started w ith low loss, moderate dielectric constant, and good lattice matching
w ith the YBCO. It m ay not be possible to find a substrate material which fits all the
requirem ents in Table 1,1; a com prom ise between different requirem ents m ay be needed
and depends on the the purposes o f applications. In this chapter, how the new substrate
m aterials are selected for study will be given in section 3.1. The basic properties and
dielectric properties at low frequencies o f these materials w ill be given too. A lthough this
thesis is focused in the m icrow ave m easurem ents and m easurem ent results, the preparation
procedure o f ceramic and single crystal samples o f new materials will be given in section
3.2 for a m ore integrated structure of this thesis.
3.1. Selection o f N ew Substrate M aterials
M aterials of perovskite structure groups are studied m ainly for the possibility as
substrate m aterials for H TSC thin films for m icrow ave applications. The first group is the
perovskite oxide family and the second group is the perovskite fluoride fam ily, as will be
discussed next.
3.1.1. A (B i B 2) 0 3 Fam ily
W akino et al. (1989), Tam ura et al. (1984), N om ura et al. (1983), and Quchi et al.
(1984) review ed and investigated a series o f ceram ic m aterials for the applications of
104
m icrow ave resonators and filters. The m aterials have very high quality factors (Q > 10,000)
as listed in part(a) o f the Table 3.1. The therm al expansion coefficients are also listed
because they are an im portant factor for the choice o f substrate too. N om ura et al. (1983)
also investigated the lattice constants and theoretical densities o f som e o f the perovskite
type m aterials as listed in part (b) o f the Table 3.1. It is expected that single crystals o f
those m aterials in Table 3.1 m ay have higher Q values than observed in ceram ics. In
addition, the Q values increase with decreasing tem perature; the Q values at liquid nitrogen
tem perature should be larger than those at room tem perature.
3 . 1 . 1 . 1 . B a (M g i/3 T a 2 /3 )C > 3
F rom T able 3.1. very high Q values are exhibited by B a(M i/ 3 M ’2 /3 ) 0 3 ty p e m aterials.
B a (M g i/ 3 T a 2 /3 )C>3 (B M T ) w as studied in this search for substrate m aterials because o f its
very high Q value, m oderate dielectric constant, and adequate lattice m atching. The
m icrow ave properties o f B M T was first reported by the N o m u ra et al. (1982) w ith
dielectric constant 2 5 and quality factor 16,800 at 10.5 GHz. M atsum oto et al. (1986) gave
a detailed study on m icrow ave dielectric properties o f BM T. H e found the dielectric
constant increases w ith increasing the relative density. H e also reported the Q values of
high density B M T (>90% ) ranged from 5,000 to 35,000 at 10 G H z and the dielectric
co n stan t by extrapolating low er densities data to 100% relative density was about 25.4. He
also found the Q value o f B M T reduces to about 2,000 w hile th e relative density drops to
low er than 85%. H ow ever, w ith relative density higher than 90% , no clear relationship
exists betw een the relative density and the quality factor. N o report o f m icrow ave
properties o f B M T at liquid nitrogen tem perature has been given yet. D ielectric properties
o f single crystal are also not available in publications. In this thesis, in addition to the study
o f m icrow ave dielectric properties o f B M T in form s o f ceram ic and single crystal at both
105
Table 3.1. Properties o f com plex perovskite com pounds at m icrow ave frequencies.
(a) Dielectric properties
M aterial
f(GH z)
£‘
Q
Tf
(ppm I ° C )
R em arks
Ref.
10.5
25
16,800
4.4
M n 1m ol %
(1)
7
25
10,200
5
10
25.4
5,000-
B a(M g]/3T a 2 /3)03
(2)
(3)
30,000
B a(M ni/3Ta2/3)O s
10.5
27.7
10,400
40
Ba(Zni/3Ta2/3)C>3
11.4
30
14,500
0.6
7
29
10,000
1
(2)
12
30
14,000
0
(4)
B a (Z n i/ 3 N b 2 / 3 ) 0 3
9.5
41
9,150
31
annealed N2
(1)
B a (Z r
10
30
10,000
0
addition o f
(2)
0 4 Z n. 3 2T a.64)03
(1)
M n 1mol 9b
(1)
Ba(Ni,Ta)Q3
B a ( N i|/ 3 N b 2 / 3 ) 0 3
(1)
(2)
(3)
(4)
(5)
10
38
11500
20
(5)
N om ura et al., 1983.
T a m u ra e ta l., 1984.
M atsum oto et al., 1986.
K aw ashim a et al., 1983.
O uchi e ta l., 1985.
(b) Structural properties (N om ura et al., 1983).
M aterial
Structure
B a (M g i/ 3 T a 2 /3 ) 0 3
hexagonal
0.5774
0.7095
7.637
B a (M n i/ 3 T a 2 /3 ) 0 3
hexagonal
0.5814
0.7156
7.709
Ba(Zni/3Ta2/3)C>3
hexagonal
0.5787
0.7087
7.944
Ba(Zni/3Nb2/3)C>3
cubic
0.4093
Lattice constant(nm)
a
j
c
D ensity
(g /cm 3)
6.515
106
room tem perature and liquid nitrogen temperature, the m icrowave properties in forms o f
pow der w ill be investigated by using the BM T:polyethylene com posite sam ples.
3.1.1.2. S r(A li/ 2 N b i/ 2 ) 0
3
, S r(A li/ 2 T a |/ 2 ) 0 3, and S r(G ai/ 2 T a i/ 2 ) 0 3
In addition to the choice o f B M T, two more new materials were investigated —
S r(A lj/ 2 N b 1/ 2 ) 0 3 (SAN ) and S r(A li/ 2 T ai/ 2 )C>3 (SA T ). The com pounds o f SA N and SAT
w ere first prepared and tested to learn their crystallographic phases and their melting
behavior by the group at the A T& T Bell Labs. (Brandle et al., 1990). C eram ic sam ples
w ere identified to have cubic perovskite structure with a/ 2 = 0.3898 nm and o f melting
tem peratures o f 1,790 °C and 1,900 °C for SAN and SA T, respectively. The Al o f the
SA T w as then replaced by G a to form Sr(G ai/ 2 T a i / 2 ) 0 3 (SGT) to com pare its properties
w ith those o f SAT after the SA T has shown very good potential to be used as the substrate
m aterial for HTSC thin films.
3.1.1.3. 0.7Sr(A l 1/2 X 1/ 2 ) 0 3 : 0.3L aA 103, 0 .7 Sr(A li/ 2 X i/ 2 )O 3 : 0.3N dG aO 3 (X = Nb, Ta)
L aA 103, as m entioned in chapter I, has good lattice m atching w ith the Y B CO
superconductor. However, it has a phase transition at ~ 435 °C which results in a highly
tw inned substrate. Recently, H aussuhl et al. (1991) and M ateika et al. (1991) reported
m ixed rear-earth alum inum perovskites by coupled substitution o f {[Sr,Ca], [T a,N b]} and
the crystal o f the m ixed com pound is twin-free w hen the cubic sym m etry is obtained. One
o f the twin free crystals they reported is the com position (La
2 8 9 Sr.7 0 9 )(AJ.6 4 6 Ta 3 5 6 )0
3
w hich m ay be considered as the solid solution o f two m em bers of LaA 103 and
Sr(Ali/2Tai/2)03.
SA N and SAT as potential substrates for H TSC, has near ideal lattice fit to the YBCO
superconductor. They also have excellent therm al expansion com patibility and desirable
107
dielectric properties, as will be described in the next section and in chapter 4. However,
both have relatively high melting tem peratures (1790 °C and 1900 °C for SA N and SAT,
respectively) which m akes the crystal growth using platinum crucible difficult by
C zochralski grow ing technique. By form ing a solid solution with a com pound o f low
m elting temperature, it is anticipated that the solid solution will result in a low er melting
tem perature. NdGaC>3 , a HTSC substrate material mentioned in chapter 1, has good lattice
m atching w ith the Y B CO and lower m elting tem perature (~ 1,500 °C), therefore was
selected as a m em ber o f solid solution with SA N and SA T for our purpose.
Five more com pounds selected for m icrowave properties testing in this thesis:
0 .7
S r(A i|/ 2 N b |/ 2 ) 0 3 - 0 . 3 L aA 1 0 3 ,
0.3N dG aO 3,
0 .3 5
0 .7
0 .7
S r(A l|/ 2 T a |/ 2 ) 0 3 - 0 .3 L aA 1 0 3 , 0.7Sr(A l j/ 2 N b i/ 2 ) 0 3 ~
Sr(A l]/ 2 T a]/ 2 ) 0 3 - 0 . 3 N dG a 0 3 , and
0 .3 5
S r(A li/ 2 N b i/ 2 ) 0 3 -
S r(A li/ 2 T aj/ 2 ) 0 3 - 0 . 3 N dG a 0 3 - T hey are to be addressed in their short-handed form as
SA N -L A , SA T-LA , SA N -N G , SA T -N G , and SA N -SA T-N G , respectively.
3.1.1.4. Basic Properties o f A (B iB 2 ) 0 3 Family
T he dielectric constant and dielectric loss o f A (B iB 2 ) 0 3 substrate materials for HTSC
at 10 kH z with different tem peratures are listed in Table 3.2 . The data w ere measured by a
G eneral Radio 1621 Capacitance M easurem ent System. It can be seen, since the dielectric
constants are expected to decrease w ith increasing frequency, all the dielectric constants are
basically acceptable as substrate m aterials o f HTSC for the applications at microwave
frequencies, as discussed in chapter 1. However, a low loss tangent at low frequency can
not guarantee a sam e order low loss tangent will happen at m icrowave frequencies;
m icrow ave m easurem ents on dielectric properties are necessary.
Properties other than complex perm ittivity o f the materials in Table 3.2 are listed in
T able 3.3. From Table 3.3, all the suggested materials have adequate lattice matching with
108
Table 3.2. Dielectric properties at 10 kH z (Guo et al., 1993a).
e'
tanS
T(°K )
22.3
2.87x10-4
300
21.9
9.0x10-5
100
23.4
3-OlxlO-5
300
23.7
< 1 .0 x l0 -5
90
(hot pressed)
24.5
6.18x10-5
300
crystal (fiber)
23.1
2.78x10*4
300
23.2
<1.0x10-5
100
23.1
1.63xl0-3
300
22.0
-1.0x10*5
100
11.89
1.68x10*3
300
11.79
4.24x10*5
100
18.64
3.18x10-3
300
18.27
9.66x10-5
100
25.68
6 .5 x 1 0 ^
300
25.7
2.79x10-4
90
21.9
3.60x10*4
300
21.7
7 .4 7 x l0 -5
90
23.26
2 .8 x l0 -3
300
23.0
5.15x10-4
90
21.06
2 .9 x l0 -3
300
-
-
90
22.8
7.80x10-3
300
22.3
5.11x10^4
90
M aterial
L aA 103
BMT
crystal
ceramic
crystal (bulk)
SA T
SA N
SA N -L A
SA T -LA
S A N -N G
SA T -N G
S A N -S A T -N G
ceramic
ceramic
ceramic
ceramic
ceramic
ceramic
ceramic
109
the Y B C O superconductor. The BMT, SA N , SA T, SAN-NG, and SA T -N G have good
therm al expansion coefficient matching with YBCO. M ost o f them are twin-free. As
expected, the m elting tem peratures o f SA N -N G and SA T-NG are low er than the SA N and
SAT, respectively. A lthough som e o f the m aterials are not perfect from some points o f
view, they are still w orthw hile for further study because o f other merits they have, as can
be seen in T able 3.3.
Table 3.3.
Som e basic properties o f some perovskite oxide m aterials o f potential
substrate o f H TSC (Guo et al., 1993a).
M aterial
Structure
Latt. const
T({ppm/0 C)
Melting
Twin
a(nm)
20°<T<300°C
point(°C )
free
0.3789
3.7
2,040
no
LaA 10 3
rhom b.
BMT
cubic
0.40877
9.0
-3 ,0 0 0
yes
SAN
cubic
0.38950
8.5
1,739
yes
SA T
cubic
0.38952
9.7
1,908
yes
S A N -L A
cubic
0.38634
_
1,705
yes
SA T -LA
cubic
0.38727
3.9
1,830
yes
S A N -N G
cubic
0.3879
1 0 .8
1,582
yes
S A T -N G
cubic
0.38866
8 .8
1,767
yes
3.1.2. L M A 1 1 O 19 Fam ily
The other oxide family suitable as substrates o f HTSC films with m icrow ave dielectric
properties to be studied in this thesis is the L M A i 1 O 19 family (L= La, Nd; M = M g, Ca;
110
A = A l, G a). T he L M A 1 1 O 1 9 family recently has been suggested for the application on
solid state laser (M erm illiod et al., 1991; C ollongue et al., 1992; V ia n a e t al., 1992).
A pplication o f one m em ber o f this fam ily (L aM gA lj 1 O 1 9 ) as the substrate o f Y B C O was
first reported by X iong et al. (1992) w ith dielectric constant 11 to 13 and loss tangent
7 x l0 - 4 at 9.4 G H z. The L a M g A ln O | 9 has a hexagonal structure w ith lattice constant a=
b= 0.5583 nm , c= 2.1935 nm . W ith the surface orientation o f (1100), the lattice m ism atch
w ith Y B C O is less than 2 % (X iong et al., 1992). Four m em bers o f this fam ily w ill be
studied in this thesis -- L aM gA li 1 O 1 9 , N d M g G aA lio O jg , C aG agA lgO ig, and C a G a ^ O ig
w ith short-handed form s as LM A , N M G A , C G A , and C G O , respectively. T he low
frequency dielectric properties and structure, and therm al expansion properties are listed in
T able 3.4. It can be seen that, in addition to good lattice m atch to the Y BCO , all the
com pounds have adequate dielectric constants for m icrow ave applications and the first
three com pounds have g o o d therm al expansion coefficient m atch w ith Y B CO .
3.1.3. K M F 3 F am ily
A nother possible potential substrate group is the perovskite fluoride fam ily K M F 3
(M = M g, M n, Zn, and Ca). T he reported dielectric and structure characteristics o f the
K M F 3 fam ily are listed in T able 3.5. F ro m the data in Table 3.5, K M F 3 fam ily is quite
attractive because o f its low dielectric constant for high speed application and good lattice
m atching w ith the Y B CO . O ne disadvantage o f the fluoride fam ily is its reaction w ith the
Y B C O thin film s. T he use o f a buffer layer m ay be required.
Table 3.4. Properties o f M L A i 1 O 19 family (G uo et al., 1993b).
(a) D ielectric constant at 10 kH z and 90 °K.
e'
tanS
LaM gA l) ) O i 9
14.0
1.51x10^
N d M g G a A lio O t 9
15.6
1.75x10^
CaGafiA leO ig
18.2
3.20x10*4
C aG ai 2
9.50
1.81x10*4
Material
0 ]9
(b) Structure and thermal expansion properties.
Material
Structure
Lattice constant (nm)
T f (ppm/°C)
a
c
300 °K
L a M g A ln O ig
hexagonal
0.5586
21.970
9.7
N d M g G a A l|o O ] 9
hexagonal
0.5894
21.918
7.2
CaGafiAlftOjg
hexagonal
0.5610
22.163
7.9
C aG ai 2
hexagonal
0.5894
20.041
~4
0
j9
112
Table 3.5. Properties o f KM F 3 family.
(a) Dielectric constant at low frequency.
M aterial
e'
Reference
K M gF3
6.97
Rittenm yer,1989
6 .0
K M nF3
Uchino, 1984
9.57
Uchino, 1984
11.1
R ittenm yer, 1989
K ZnF3
7.9
Rittenm yer, 1989
K C aF 3
8.5
Rittenm yer, 1989
(b) Structure and thermal expansion properties.
Material
Structure
Lattice constant(nm)
a
K M gF3
K M nF 3
K ZnF3
K C aF 3
TKPpm/°C)
b
c
<200°C
-5 5 0 °C
18.74
21.95
Reference
cubic
0.3988
-
-
G uo, 1993
cubic
0.4182
-
_
,
-
cubic
0.4184
-
-
14.3
11.5
cubic
0.3973
-
-
-
ortho
0.8095
0.8110
0.4049
16.3
27.9
cubic
0.4055
-
-
-
-
G allasso, 1969
ortho
0.6166
0.6207
0.4382
-
-
G uo, 1993
cubic
0.8742
-
-
-
-
Gallasso, 1969
G allasso, 1969
G uo, 1993
G allasso, 1969
G uo, 1993
i 13
3.2. Sam ple Preparation
B oth ceram ic and single crystal sam ples were prepared by a group in the IM R L o f
P enn State. The ceram ic sam ples w ere also used for the ceram ic seed and feed rod for
single crystal fiber grow th, as w ill be m entioned later.
3.2.1. C eram ic S am ples
C eram ic sam ples were prepared by solid state reaction, using conventional techniques.
Several batches w ere prepared under various calcining and sintering conditions and the
procedure w as optim ized through processing study. Figure 3.1 is the flow chart o f the
preparation procedure. Som e o f the processing param eters used are sum m arized in T able
3.6 (G uo et al., 1993a).
3.2.2. L aser H eated Pedestal G row th
T he laser heated pedestal grow th (LH PG ) m ethod has been show n to be a pow erful
m ethod for rapidly grow ing sm all diam eter single ciystal, particularly, o xides o f high
m elting tem perature, for both property study and fiber devices (Feigelson, 1988). The
L H P G equipm ent consists o f a p ow er source (w ater cooled, tunable flow ing gas C O 2 55W
laser), an optical layout, and a grow th section. The circular laser radiation o f TEM oo m ode
w as transform ed into an annulus by a reflaxicon (Saifi et al., 1986). T he annulus was
directed onto a parabolic m irror that focused the radiation back to its focal point, form ing
the hot zone. T he pulling heads are high precision m icrostepper m otors w ith a single step
increm ent o f 25 p m . The grow th ch am b er is a stainless steel vacuum ch am b er enclosed the
reflaxicon and internal optics to m inim ize air current disturbance and allow grow th to take
place in a reducing or oxidizing atm osphere. T he alignm ent, seeding, and g row th are
Bail M ill C arbonates and Oxides
D ry and Calcine
G rind and Sieve
Add B inder. G rind, Sieve and Press Pellet
B um o u t B inder
Sinter
Slice W afers and Polish the Surface
Cut O ut Preform s
C lean, R eady for G row th Test
Figure 3.1. Processing procedure o f the ceramic preform .
115
Table 3.6. C eram ic processing conditions (Guo et al., 1993b).
C om position
BMT
Calcine
Sinter
1500 °C
1670 °C
for 3 hours
for 3 hours
Starting chem icals
BaCC >3 (G rade 1)
M gO (3N5)
SA N
1570 °C
1600 °C
for 3 hours
for 3 hours
T a2 0
5
(4N)
S r C 0 3 (4N)
SA T
1600 °C
1620 °C
for 3 hours
for 3 hours
A12 0
N b2 0
SA N -L A
SA T -LA
S A N -N G
S A T -N G
1500 °C
1500 °C
for 3 hours
for 3 hours
1500 °C
1625 °C
for 3 hours
for 3 hours
1575 °C
1575 °C
for 3 hours
for 3 hours
1575 °C
1575 °C
for 3 hours
for 3 hours
3
5
(3N5)
(4N)
L a(O H ) 3 (3N)
116
visually imaged outside the growth cham ber by a short focal length telescope. M olten zone
tem perature during a stable growth was monitored using an optical pyrom eter with
resolution o f linear dim ension o f 0.1 mm . A schem atic diagram o f the LH PG station is
show n in Figure 3.2 (G uo et al., 1993). Further detailed description can be found elsew here
(Y am am oto, 1991).
117
Discharge
Current
Control
Telescope
Growth Chamber
Optic Pyrometer
■■■■wf
I
IVIV
Beam Staerer
Carbon
Dioxide
Laser
Beam Selector
Power Sensor
Power M eter
Beam Splitter
Mirror
Figure 3.2. Schematic diagram of the laser heated pedestal growth station.
118
Chapter 4
E X PE R IM E N T A L R ESU L TS AND D ISC U SSIO N S
This chapter gives the dielectric m easurem ent results by different m easurem ent
techniques for the substrate materials for HTSC thin films at microwave frequencies. The
m easured dielectric properties of some used substrate m aterials for the HTSC thin film
substrates are given in section 4 .1. The results o f new substrate materials o f the oxide
fam ily are described in section 4.2. The section 4.3 is the m easurem ent results o f the
fluoride family. One cavity perturbation and four dielectric resonance techniques are used
in the m easurem ents o f this chapter. The cavity perturbation technique is used to measure
the dielectric properties o f small pieces or fiber samples which can not be m easured by the
dielectric resonance techniques because o f sample dim ensions required by those methods,
as discussed in chapter 2. Sam ples with large enough dim ensions are m easured m ainly by
the dielectric resonance techniques because o f their higher accuracy on loss measurement.
Som e m aterials are m easured by more than one techniques to com pare and exam ine
accuracy o f the results by different techniques. The accuracy of the m odified model o f the
parallel plate resonance m ethod suggested in chapter 2 will also be discussed in this chapter
by com paring the results obtained from different techniques. The accuracy o f the dielectric
properties/frequency relationship by equation (1.8) is also exam ined. To keep the
consistence, all the dielectric properties calculations (including those o f the cylinder cavity
and the w aveguide reflection resonance techniques where com puter program s to deal with
the sidew alls are not available) in this chapter are conducted by the modified model o f the
parallel plate resonance technique unless the resonant frequency is over the cutoff condition
(8.82 G H z and 6.77 G H z for the cylinder cavity technique and the reflection technique with
119
C band cavity, respectively), i.e., the height o f the cavity is larger than one half wavelength,
w here the D RESV 2 program will be used.
4.1. M easurem ent R esults o f Used M aterials
T he AI 2 O 3 single crystal fiber and the L aA 1 0 3 bulk single crystal were m easured by
the cavity perturbation technique at the X band and the K band frequency regions at room
tem perature. The m easured dielectric properties are listed in Table 4.1. The dielectric
constants and the losses w ere calculated by the equation (2.35). Som e o f the dielectric
losses are low er than the limitation o f the cavity perturbation technique on the
m easurem ents o f low loss sample, as m entioned in chapter 2. L ow er limits o f the Q
factors are judged based on the fluctuations in measurem ent results.
Table 4.1. Dielectric properties o f AI2 O 3 single crystal fiber and LaA lO s single crystal.
LaA lQ 3
AI 2 O 3
M aterial
8.5877
10.1576
11.8367
19.1613
21.2096
11.95
11.73
11.84
20.44
21.49
tanS
< 5x10*4
< 5 x l 0 -4
< 5 x l0 4
<5x10*4
6.28x10-4
Q
>
>
>
>
fc (G H z)
e’
2 ,0 0 0
2 ,0 0 0
2 ,0 0 0
2 ,0 0 0
1,590
F o r the AI2 O 3 fiber, the electric field was along the extraordinary axis, therefore only
the dielectric properties along the extraordinary direction w ere measured. The m easured
dielectric constants are quite close to the value reported by the Bystrov et al. (1989) in Table
120
1.2. For the m easurem ents on the LaAlC >3 single crystal, the dielectric properties are
m easured at K band frequencies instead o f X band frequencies because the available
sam ple length is too short to be m ounted in the X band cavity. The dielectric constants o f
LaAlC >3 are consistent w ith the low frequency datum in Table 3.4. H ow ever, two different
dielectric constant values o f LaA 1 0 3 w ere reported, as listed in the Table 1,2. By using
fxQ = constant (W akino et al., 1986), the dielectric loss data fall betw een the datum
reported by the Sim on et al. (1988) and that reported by the Konaka et al. (1991), in Table
1.2. H ow ever, no confirm ation shows the expression fxQ= constant can be correctly
applied to all the materials at microwave frequencies,
4.2. M easurem ent Results o f the Oxide Family
B oth perturbation technique and dielectric resonance techniques w ere used to m easure
the m aterials o f the oxide family depending on the dielectric constant and the available
sam ple geom etry. For the dielectric resonance techniques, the conductivity o f metal shields
is required for the com putation o f conductor loss, especially for the post resonance
technique, as discussed in chapter 2. The m easurem ent o f the conductivity o f the brass
shields for the post resonance method will be given first. The m easurem ent results on
ceram ic sam ples by different techniques other than the parallel plate resonance technique
will be given in sections 4.2.2. The section 4.2.3 gives the measured results by the parallel
plate resonance m ethod. Dielectric constant data from single crystal fibers measured by the
cavity perturbation technique will be given in section 4.2.4.
121
4.2.1. M easurem ent o f Conductivity o f M etal Shields
As m entioned in chapter 2, the m easurem ent o f surface resistance can be conducted by
using the setup o f post resonance method. Either tw o sam ples w ith the thickness o f one
sam ple N tim es that of the other sam ple (where N accounts fo r the N of T E o in m odes) or
one sam ple w ith the square o f diam eter/thickness ratio close to 3 can be used for
m easurem ent by using equations (2.3) or (2.4), respectively. In this thesis, by using the
equation (2.3), tw o alum inum sam ples w ith diam eter 12.6 m m and thicknesses 7 m m and
14 m m , respectively, are used. The alum inum sam ple with diam eter/thickness equal to
12.6 m m /7 m m is also adopted for the Rs m easurem ent o f using equation (2.4). The
m easurem ent results on the surface resistance o f the brass shield are then converted to the
conductivity by using equation (2 .2 0 as given in Table 4.2. A 3 % error was found on both
m easurem ents using equations (2.3) and (2.4), w hich m eans, from equation (2.26), a 1%
to 9 % error on loss tangent m easurem ent can happen by the post resonance technique for
m aterials w ith Q factor 1,000 to 10,000.
Table 4.2.
Param eters o f the shield conductivity m easurem ents.
M ode
Frequency (GHz)
R s (Q)
a ( 1 /Q -m )
(2.4)
(2.3)
Equation
TEon
TE0 1 2
9.786
TE0 1 2
TE02 t
15.576
15.344
0.05214
0.06596
1 .4 2 1 x l0 7
1.405x107
122
It is surprising that very good agreem ent was found betw een the m easurem ents o f
conductivity using the tw o different methods. The average conductivity value 1 .4 lx l0 7 /H m by the two m ethods was then used for the m easurem ents by the post resonance and
parallel plate resonance techniques. For the cylindrical cavity and the w aveguide reflection
techniques, where shields are made o f copper, reported Cu conductivity 5 .8 x l0 7 /Q -m is
used for the calculation o f conductor loss o f copper shields in the two m ethods. The
conductivity o f copper shields used may not be exactly the sam e as the reported 5 .8 x l0 7
/H -m because the shield surface may be influenced by oxidation, scratch, and other factors.
H ow ever, the error on dielectric loss m easurem ent due to that conductivity error is
negligible because o f the low conductor losses o f both methods.
4.2.2. M easured R esults o f Ceramic Sam ples
The dielectric properties at room temperature o f the potential substrate materials for
H TSC films with ceram ic form at m icrow ave frequencies are listed in Table 4.3.
M easurem ents were conducted using all different dielectric resonance techniques except the
parallel plate resonance technique, which will be given individually later. Equations (2.1),
(2.2), (2.5), and (2.6) w ere used for calculations o f dielectric properties o f various
techniques. Conductor losses on the sidew all were neglected because o f the adequate
com puter program was not available and the conductor losses were expected to be very
low on these two techniques. Some o f the materials with dielectric data already listed in
Table 4.3 were chosen to be measured by the perturbation method to confirm the accuracy
o f dielectric constant m easurem ents, as shown in Table 4.4. Since the disk shape sample of
S A T-LA was not available, it was only m easured by the cavity perturbation m ethod with
results also shown in T able 4.4. The loss data o f some frequency points o f the perturbation
m ethod are not listed because o f the instability o f the results or the limit o f the technique for
123
Table 4.3.
M icrow ave dielectric properties at ro o m te m p e ra tu re o f the oxide fam ily o f
the potential substrate materials for H TSC thin films at c e ra m ic form.
Sam ples were m easured by dielectric resonance techniques.
Material
Technique*
Frequency
£’
tan5
Q
(GHz)
BMT
BMT
(hot pressed)
SA N
SA T
SG T
NMGA
P.R.
1 0 .8 6
23.49
1.646xl0*4
6,077
C.C.R.
6.551
-
8 .3 4 4 x l0 *5
11,985
W .R.R .
6.390
*
8.104x10-5
12,339
P.R.
17.85
24.14
4.127x10-4
2,423
C .C.R
1 0 .2 0
.
1.982x10-4
5,044
P.R.
8.560
1.035x10-3
967
C.C.R.
6.085
-
9.804x10-4
1 ,0 2 0
W .R.R .
5.890
-
9.074x10-4
1 ,1 0 2
P.R.
11.03
10.70
5.969x10-4
1,675
C.C.R.
7.644
-
4.024x10*4
2,485
W .R.R .
7.553
.
3.869x10-4
2,584
P.R.
10.59
26.77
2.412x10-4
4,146
C.C.R.
6.501
-
1.350x10*4
7,409
W .R.R .
6.332
1.325x10*4
7,549
P.R.
17.79
6.819x10-4
1,467
C.C.R.
11.13
-
4.877x10*4
2,051
W .R.R .
11.58
-
5.262x10-4
1,900
Table 4.3 continued by next page
18.25
13.70
124
Continued from the form er page.
M aterial
Technique
Frequency
e'
tan5
Q
(GHz)
S A N -L A
S A N -N G
S A T -N G
S A N -S A T -N G
*
P.R.
8.058
24.54
3.3 9 2 x l0 -4
2,948
C.C.R.
5.451
-
2 .5 7 7 x l0 *4
3,881
W .R .R .
5.311
-
2.503x10*4
3,996
P.R.
9.681
24.83
1.020x10*3
980
C.C.R.
5.832
-
7.220x10-4
1,385
W .R .R .
5.688
-
6.930x10*4
1,443
P.R.
10.51
20.70
5.636x10-4
1,774
C.C.R.
6.372
3.957x10-4
2,527
W .R .R .
6.149
-
3.791x10-4
2,638
P.R.
11.26
2 1 .2 0
7.922x10-4
1,262
C.C.R.
6.538
-
6.369x10*4
1,570
W .R.R .
6.299
-
5.602x10-4
1,785
P.R.: Post resonance technique;
C.C.R.: Cylinder cavity resonance technique;
W .R.R .: W aveguide reflection resonance technique.
125
Table 4.4.
M icrow ave dielectric properties at ro o m te m p e ra tu re o f the A(B iB 2 )C>3
family o f the potential substrate m aterials for HTSC thin film s at ceram ic
form . Sam ples w ere m easured by the c a v ity p e r tu r b a tio n te c h n iq u e .
Material
fc
(G H z)
£’
tan5
(xlO-3)
Q
BM T
8.5894
23.07
<0.5
>2 ,0 0 0
10.162
2 2 .8 6
<0.5
>2 ,0 0 0
11.938
23.36
<0.5
>2 ,0 0 0
8.5914
18.52
-
-
10.161
18.12
-
-
11.936
18.80
-
-
8.5894
24.37
1 .1 2
889
10.159
24.28
0.997
1,003
11.934
24.65
2.04
490
8.5894
20.41
<0.5
>2 ,0 0 0
10.159
20.92
<0.5
>2 ,0 0 0
11.934
20.71
<0.5
>2 ,0 0 0
8.5891
21.30
<0.5
> 2 ,0 0 0
10.159
22.32
<0.5
>2 ,0 0 0
11.933
22.37
<0.5
>2 ,0 0 0
SA N
SA N NG
SA TNG
SA TLA
126
m easuring low loss sam ples. The dielectric properties o f the L M A 1 1 O 19 fam ily are listed
in T able 4.5. T he dielectric loss data at liquid nitrogen tem perature m easured by the
cylinder cavity resonance m ethod are given in Table 4.6.
A s seen in T able 4.3, there is good agreem ent on the m easured quality factors using the
cy lin d er cavity resonance m ethod and the w aveguide reflection resonance m ethod. Since
the loss data m ay change w ith the frequency, the m easured frequencies o f the post
resonance technique are far from those o f the cylinder cavity and the reflection resonance
techniques fo r direct com parison o f Q factor m easurem ents. B asically, Q factors decrease
w ith increasing the frequency, as is reasonable referring to equation (1.8). B y com paring
the data o f T ables 4.3, 4.4, and 4.5, good agreem ent is also found on the m easurem ents o f
dielectric constants betw een the post resonance m ethod and the cavity perturbation
technique. T herefore, the accurate m easurem ents o f dielectric constant and quality factor are
dem onstrated. It can also be seen that the change o f dielectric constant w ith frequency at
m icrow ave frequencies is negligible thus confirm s the result o f equation ( 1 .8 a).
B y repeating the Q factor m easurem ents, a 2 to 5% , -2 % , and 3 to 6 % m easurem ent
errors can be found for the post, cylinder cavity, and w aveguide reflection resonance
techniques, respectively. Those errors are higher than the theoretically d iscussed values in
chapter 2. F o r the post resonance m ethod, tw o situations m ay cause the h ig h er error. The
change o f surface resistance due to the surrounding environm ental conditions (tem perature,
hum idity, oxid atio n o f m etal shields) m ay be higher than
1
% error adopted fo r discussion
in chapter 2. T he other possible error source is the uncertainty error (E q 7 ). T h e 2 % error
for the cylinder cavity technique is reasonable w hen com pared to the theoretical value
w here total erro r is about 1% for E q i to E q 4 , as discussed in chapter 2. T he 1% difference
should com e from the uncertainty erro r too. The high error o f the w aveguide reflection
m ethod is from the uncertainty error in p o w er level instability. T he resonance trough was
127
Table 4.5.
D ie le c tric c o n sta n ts* o f the LM A i jO jg fam ily in ceram ic form m easured
by the c a v ity p e r tu r b a tio n te c h n iq u e .
Frequency (GHz)
8.5895
10.1592
11.9335
LM A
10.94
10.99
11.31
NMGA
13.78
13.80
14.4
CGA
17.45
18.01
18.94
CGO
9.10
9.20
9.38
* A ll the Q > 1,000.
Precise m easurem ents on Q factors are not available.
T able 4.6.
M icrow ave loss data at liq u id N 2 te m p e ra tu re o f the oxide fam ily o f the
potential substrate m aterials for HTSC thin films at c eram ic form . Sam ple
are m easured by the c y lin d e r c a v ity re s o n a n c e te c h n iq u e .
Frequency (GHz)
tanS
Q
BMT
6.561
2 .9 8 4 x l0 '5
33,512
SA N
6.076
5 .9 0 3 x l0 '4
1,694
SA T
7.632
2.503xl0-4
3,996
SG T
6.535
1.276x10-4
7,834
NMGA
11.13
3.973x10-4
2,517
SA N -L A
5.485
1.603x10*4
6,239
S A N -N G
5.897
5.974x10-4
1,674
S A T -N G
6.394
3.556x10*4
2,812
S A N -S A T -N G
6.564
4.496x10-4
2,224
M aterial
128
very sensitive to the positions o f the sam ple and shorted end thus causing the high erro r in
reproducibility. O nly from the point o f view o f m easurem ent erro r on Q factor
m easurem ents, the cylinder cavity resonance technique is the best m ethod am ong the above
three techniques. It has to be pointed out is that som e error sources can not be observed by
repeating m easurem ents because they do not change w ith m easurem ents, for exam ple, the
calculation errors o f conductor & radiation losses and factor "A". In addition, the
m easurem ent error on the shield surface resistance can not be seen either by repeating
m easurem ents. T he 1% to 9% m easurem ent error m entioned in section 4.2.1 for the post
resonance m ethod is an exam ple. By adding this error to the erro r observed by repeating
m easurem ents, the total error on loss tangent m easurem ent for the post resonance
technique w ill be 2% (Q = 1,000) to 8 % (Q = 6,000).
T he ultra low loss and adequate dielectric constant o f the B M T m ake it a good choice
fo r the H TSC substrate m aterial for applications requiring low loss, e.g., for the application
o f m icrow ave antenna. The lattice m ism atch o f B M T w ith the Y B C O m ay be a little too
large ( 6 % ) fo r the application, and its high m elting tem perature is considered a huge
hindrance to the grow th o f large single crystal. H ow ever, for the m icrow ave antenna type
application, a single crystal fiber w ith diam eter in the order o f m m is enough and can be
gro w n well by the LH PG technique m entioned in chapter 3 (G uo et al., 1993c). T he SG T
and S A N -LA have loss tangents close to the requirem ent for antenna application at liquid
nitrogen tem perature. Loss tangents o f single crystal form s are expected to be further
low er. T heir dielectric constants are also suitable for the applications requiring higher
dielectric constant values.
A lthough the loss tangent o f SA T m ay be a little too high fo r antenna application, it has
a low dielectric constant com parable to Si or other sem iconductors w hich m akes it a very
strong potential substrate m aterial for devices requiring high speed or m ultilayer design
129
w hich usually do not require a very low loss tangent. It also has good lattice and thermal
expansion m atches with the Y BCO superconductor. M ost o f the L M A 1 1 O 19 fam ily have
show n low dielectric constant values. The LMA and CG O have dielectric constants
com parable to SAT. The loss tangents o f LMA, CG A, and CG O need to be further
studied. The N M G A has higher dielectric constant values than SAT, how ever, its dielectric
constant is still considered to be quite attractive for high speed and m ultilayer applications
in com parison to other higher dielectric constant substrate materials. The loss tangent o f
N M G A at a sam e frequency point is low er than that o f SAT if considering that the loss
tangent w ill decrease with decreasing frequency at m icrowave frequencies.
The dielectric constants o f SA N -N G , SA T-LA , SA T-N G , and SA N -SA T-N G are
good for applications requiring higher dielectric constants. The SA N -N G has the lowest
m elting tem perature for crystal grow th but with the low est Q factor. The dielectric
constants o f SAN and CGA are between the two types applications, but are m ore adequate
for the applications requiring higher dielectric constant values. Loss tangents o f the above
six m aterials are too higher to be applied at m icrow ave antenna, but are low enough for
other applications.
There are two more points which have to be noticed.for the above m aterials. First is
that the lattice m ism atch o f N M G A and CG O w ith Y BCO is higher than that o f BM T, as
can be seen in Table 3.4. The other is, from Tables 3.3 and 3.4, the therm al expansion
m ism atch o f SA T -LA and CGO with Y BCO is quite high.
4.2.3. Frequency D ependence o f Dielectric Properties
A s m entioned, the dielectric constants in Table 4.3 are alm ost independent o f
frequency in the microwave frequency range and agree with the expression o f equation
(1.8). T he relationship o f frequency and the loss tangent of the perovskite oxide family
130
were exam ined by the parallel plate resonance technique. The results are given in Figure
4.1. The distances betw een the sample and the metal shields w ere changed for
m easurem ents o f different frequencies. Basically, the quality factors decrease w ith
increasing frequency. The quality factors m easured by the cylinder cavity m ethod are also
show n in Figure 4.1 as solid dots for com parison. G ood agreem ent can be found in the
m easurem ent o f quality factors between the parallel plate and cylinder cavity resonance
techniques. T he Figure 4.2 show s the dielectric constant data m easured by the parallel plate
method, w hich are divided into parts (a) and (b) for a clearer view. The dielectric constants
m easured by the post resonance technique are show n by dashed lines for com parison by
assum ing the dielectric constant is frequency independent. The low er dielectric constants at
low er frequencies (larger distance between sam ple and shields) occur because o f error in
predicting the resonant frequency o f the m odified field model as show n in Figure 2.12, and
are not characteristic o f m aterials themselves. G ood agreement o n the m easurem ent o f
dielectric constants by the parallel plate and the post resonance techniques was also
obtained. Therefore, accurate measurements on both dielectric constants and quality factors
are further confirm ed. The error on Q factor m easurem ents in repeating the experim ents is
about 2% to 5 % depending on the quality factor o f sample and the distance M betw een the
sam ple and the shields. F o r M close to zero the erro r is close to that o f the post resonance
m ethod. By increasing the M value, the error from repeating m easurem ents is close to 2%,
the sam e as the cylinder cavity technique. A t this point, the radiation loss becom es more
im portant, b u t the calculation error on Qr can not be seen by repeating measurements.
Figure 4.3 is the frequency dependence o f fxQ using the d ata in Figure 4.1. A s can be
seen, the fxQ is constant for B M T and SAT, agreeing with the equation (1.8). For other
m aterials, slow increases happen on fxQ w ith increasing frequency. However, since the
slopes are quite small, w ith the exception o f SA N , it can still be concluded that equation
O
BMT
□
SGT
o
SAN-LA
SAT-NG
SAT
NMGA
SANSAT-NG
SAN-NG
SAN
A
AO
O
□
o
A
O
9
11
13
15
Frequency (GHz)
Figure 4.1.
Quality factors of perovskite oxide family measured by the parallel plate resonance technique.
The solid dots are the results of cylinder cavity resonance method for comparsion.
B
A
BMT
SGT
A
•
SAN-NG
SAT-NG
♦
SAN
----------Post resonance technique.
A
A
r - ____________ __
A .
a
_______ A
__________
■■
i
5
I_____i
6
SAN-NG
A
A
A
.......... ■ . . .
■ ------------------------------
_____ ■_
B
SGT
A
_
BMT
________
---M------
___
•
^ SAT-NG
- A ------------------ •
-----------
•
!____ i
7
1___ i
8
1___ i
9
1----- 1--------- 1---------1-----
10
Frequency (GHz)
Figure 4.2.
Continued on next page
11
12
30
Dielectric
constant
25
:
A
NMGA
e
sat
h
▲
+
SAN-LA
---------- Post resonance technique.
+ _-t_
S A N -L A _ _
:+
---------------------------
4 ------------- A ---------------------------
▲
20
15
SAN-SAT-NG
SAN-SAT-NG
r_ .NMGA_________________________
A___A .
K
10
SAT
5
5
(b)
Figure 4.2.
7
9
11
13
15
Frequency (GHz)
Dielectric constants of perovskite oxide family measured by the parallel plate resonance
technique. Dashed lines are the results from the post resonance technique.
BMT
SGT
f (GHz)
SAN-LA
NMGA
SAT
SAT-NG
SAN-SAT-NG
SAN-NG
SAN
5
7
9
11
13
Frequency (GHz)
Figure 4.3.
Relationship o f fxQ and frequency o f perovskite oxide family.
15
135
( 1 .8 b) roughly describes the loss tangent/frequency relationship o f the perovskite oxide
family w ith frequency around the order o f several gigahertz that is the loss tangent is
proportional to the frequency.
4.2.4. M easured Results o f Single Crystal Fiber Samples
T he m easurem ent data o f single crystal fibers by the cavity perturbation technique are
listed in Table 4.7. The loss tangent data are not m easured because o f the loss m easurem ent
lim itation o f the perturbation method. By comparison, the dielectric constants o f single
crystal fibers are higher than those o f the ceram ic materials. That phenom enon is
reasonable because the relative density o f single crystal should be higher than that o f the
ceram ic, and the dielectric constant usually increases with increasing the relative density
(M atsum oto et al., 1986). How ever, for some materials, the differences on the dielectric
Table 4.7.
D ie le c tric c o n s ta n ts o f som e perovskite sin g le c r y s ta l f ib e r s m easured
by the c a v ity p e r tu r b a ti o n te c h n iq u e .
fc (G H z)
BM T
SA T
SA N
S A N -N G
8.586
26.94
_
.
28.41
24.72
10.16
26.77
_
.
28.83
25.31
11.93
27.43
_
.
30.19
_
25.30
19.16
•
18.13
2 1 .8 8
_
21.47
-
21.21
-
18.66
21.96
-
22.40
SA T-LA
SA T -N G
136
constants o f single crystal and ceramic are too large to be explained by this density
differences. T his m ay com e from the error on sample volume m easurem ent o r the surface
o f the fibers was not sm ooth enough for accurate m easurements.
4.3. M easured Results o f the KM F 3 Family
The dielectric properties of the single crystal o f KM gF 3 and the ceramic o f K Z 11F 3 are
m easured and listed in Table 4.8. M easurem ents by the waveguide reflection technique
were not conducted because the interferences o f exterior and mixed m odes at high
frequencies deteriorate the accuracy on pow er level measurements. The parallel plate
technique w as also not used because o f the high radiation losses on low dielectric
constants. It w as found the K M gF 3 has a very high quality factor and a low dielectric
constant w hich making it a very attractive substrate material for H TSC thin films for
applications in microwave antenna, high speed, and m ultilayer design. Its reaction with the
thin film is the m ain disadvantage o f this material. The thermal expansion m ism atch o f
K M F 3 fam ily with Y BCO is quite large, as can be seen from Table 3.5, which is the other
disadvantage. T he K Z 11F 3 also has a low dielectric constant and adequate loss tangent as
the H TSC substrate material. However, it has a higher lattice m ism atch with the YBCO
than that o f K M gF 3 . T he accuracy of dielectric constant is also confirm ed by m easuring the
dielectric constant o f K M gF 3 at K band frequencies.
4.4. Sum m ary
T he m icrow ave dielectric properties of potential substrate materials of both perovskite
oxide fam ily and perovskite fluoride family for high Tc superconductor thin films have
been m easured. M ost o f the materials have show n very good potential to be used as the
137
Table 4.8.
Dielectric properties o f the KM F 3 family.
(a) Sam ples measured by the dielectric resonance techniques.
Material
K Z n F 3 ++
K M gF 3 +
C.C.R.
C.C.R.
P.R.
Technique*
P.R.
Tem perature
300°K
300°K
77°K
300°K
300°K
77°K
Freq. (G H z)
20.533
16.859
16.890
22.918
17.382
17.362
6.19
-
-
8 .0 0
-
-
e'
tano
0
1.16x10^
1.42X10-4
7.08xl0*5
7.32x10*4
3.52x10-4
3.07x10-4
8,641
9,521
19,118
1,366
2,838
3,257
* P.R.: Post resonance technique;
C.C.R.: C ylinder cavity resonance technique.
+ Single crystal.
+ + Ceram ic.
(b) K M gF 3 measured by the cavity perturbation technique.
fc
(G H z)
19.160
5.94
21.206
6.06
138
substrate o f H TSC thin films for various applications. For the application o f m icrow ave
antenna, the B M T is the top choice and SG T and SAN-LA can also be good choices. The
SAT is the best candidate for the applications requiring low dielectric constant. The LM A ,
C G O , and N M G A also show good potential. The SGT, SA N -L A , SA N -N G , SA T -LA ,
SA T-N G , SA N -SA T-N G , SA N , and C G A are good for application requiring higher
dielectric constant. The K M F 3 family have show n very low dielectric constants. The low
dielectric constant and loss o f KM gF 3 is good for the application o f antenna, high speed,
and m ultilayer design.
The accuracy of different microwave m easurem ent techniques has been confirm ed.
The quality factor/frequency dependence at microwave frequencies has been exam ined in
this chapter as well. It was found the fxQ= constant relation is only roughly correct fo r the
perovskite oxide family.
139
Chapter 5
E S T IM A T IO N O F T H E D IE LEC TR IC PR O PER TIES O F B a(M gi/ 3 T a M ) 0 3
B Y PO W D ER M IX IN G M ETH O D
5.1. Introduction
From the study in the form er chapters, the B a(M gi/ 3 T a 2/3 ) 0 3 is a good candidate for
the substrate material for the superconductor thin films. The dielectric properties o f
B a(M gi/ 3 T a 2 /3 ) 0 3 ceramic and single crystal fiber at m icrow ave frequencies have been
studied in chapter 4. In this chapter, the pow der m ixing m ethod is used to predict the
dielectric properties o f B a(M gi/ 3 Ta 2/ 3 ) 0
3
ceram ic at m icrow ave frequencies by
extrapolating the dielectric properties of various com posite com pounds. The polyethylene
(PE) w ith density 0.915 g/cm 3 was chosen as the m atrix material w ith reported dielectric
constant 2.25 and tan5= 4xICH at 10 G H z (Pozar, 1990, p. 715). The BM T po w d er was
m ixed w ith the PE pow der to m ake 0:3 com posite samples w ith different sizes and shapes
for the m easurem ents o f various m icrow ave m easurem ent techniques at X band
frequencies. Three different m icrow ave m easurem ent techniques w ere used for the
m easurem ents o f the dielectric properties o f m ixture samples in this chapter. They are the
S 1 1 and S21 technique, the cavity perturbation technique, and the post resonance technique.
For the first tw o techniques, specim ens are cut from the same sam ple for testing. Sam ples
for the post resonance m ethod are made separately.
140
5.2. Sam ple Preparation
Suitable am ounts o f PE and BM T powders w ere m ixed using alcohol as a solvent and
ZrOa balls in a plastic ja r by ball milling m ore than 10 hours. A fter the well m ixed
suspension was stirred and evaporated until alm ost dry, it was baked at 80 °C for 4 to 5
hours to obtain the com plete dried pow der m ixture. The mixed pow der was then pressed
using a steel die at tem perature 125 °C (the m elting point of PE is 115 °C) and under a
pressure -7 ,0 0 0 lb/in 2 to m ake slab samples.
5.3. M ixture Rules
A series o f studies on the m ixture rules are available in the literature (Tinga et ah,
1973). T he logarithmic m ixture rule,
log £’ -
VjlogEj
+
V 2 log
e
(5 . 1 )
2
and M axw ell’s mixture rule,
v2m !3____
+—
>+ vi£.
3 e 2____________
£
=
V 2( | + —
3
3
) + V,
(52)
e2
are tw o o f the m ost popular m ixing rules, where e',
E i,
and
£2
are dielectric constants of
com posite, dispersed particles and m atrix and V j and V 2 are the volum e fractions o f
materials w ith dielectric constant £j and £2 , respectively. The logarithm ic mixture rule
applies well in the case o f high dielectric constant phase dispersed in a low dielectric
constant m atrix which is adequate in the present case o f BM T dispersed in the PE matrix.
141
In addition, the logarithmic m ixture has the advantage o f a very sim ple relation for deriving
the extrapolated dielectric constant values from the measured results. The M axw ell’s
m ixture rule usually yields good agreem ent with m easurem ent w hen the volum e
percentage o f disperse particles is small. Recently, W akino et al. (1993) reported a new
m ixing equation,
( V , - V 0)
£’
< V, - v „ >
= V 1£ 1
( V , - V 0)
+ V 2e2
(5.3)
w here the Vo is an em pirical value about 0.35. For V i =V q, the equation is the sam e as the
logarithm ic mixture rule. T hey claim ed the new equation is the best among equations thus
far reported.
In this chapter, the logarithm ic, the M axw ell’s, and the new m ixture rules are used to
com pare w ith our m easured dielectric constant values o f com posite samples. The predicted
dielectric constant o f B M T by extrapolating the dielectric constants o f various composite
com pounds using the logarithm ic rule is also given.
B ecause the surface conditions (roughness and porosity) o f the com posite sam ples
could not be precisely controlled, the quality factors ( Q - 1/tan5) o f the sam ples are not
precise. Estim ation o f Q values by the pow der m ixing m ethod is not adequate w ithout
solving the surface condition problem s, as will be proved by the experim ental results o f
this chapter.
5.4. R esults and D iscussions
T he dielectric properties measured by different m icrowave techniques w ith different
volum e loading percentages are listed in Table 5.1. The dielectric loss data m easured by the
S 11 and S21 m ethod are not listed in Table 5.1 because of the lim itations o f loss
142
Table 5.1. Dielectric properties o f B M T-PE com posite samples.
(a) Dielectric Constant:
V olum e %
S l l & S21
£’
Perturbation
100% PE
2.25
2.29
2.32
9.05% BM T
3.09
3.16
3.28
19.3% B M T
4.10
4.01
4.39
37.5% B M T
6.09
5.91
5.82
58.1% B M T
8.53
8.48
-
(b) Q uality Factor at 10 GHz:
Q
100% PE
Perturbation
Post Resonance
-
4,218
9.05% B M T
1,160
1,010
19.3% B M T
630
700
37.5% B M T
210
921
58.1% B M T
180
-
Post Resonance
143
m easurem ent by the transm ission technique. For the S 11 and S21 and the perturbation
techniques, the tested specim ens for different loading percentages were cut from the same
sam ple pieces for easier com parison o f the results o f the tw o methods. D ata from the S I 1
and S21 technique are the average values m easured from 8.2 G H z to 12.4 G H z in Figure
5.1. D ata from the perturbation technique were m easured at frequency 10 G H z. Com paring
the results o f the three different techniques, good agreem ent on the dielectric constant
m easurem ent was obtained. In addition, the measured dielectric constant values o f
polyethylene agree well with the reported value o f 2.25. The accuracy of dielectric constant
m easurem ent by the S I 1 and S21 technique is proved. H ow ever, the m easured Q factor o f
PE at 10 G H z is higher than the reported value o f 2,500.
In Figure 5.2, the m easured dielectric constants by the three methods are com pared to
three different mixture rules. It was found that the m easured data roughly fit all the mixture
rules. It is surprising that the sim plest logarithmic mixture rule seem s to fit the m easured
data best. The extrapolated dielectric constant values by the logarithmic rule are about 23,
22.5, and 27 for the perturbation, the transm ission, and the post resonance techniques, as
show n in the Figs. 5.3, 5.4, and 5.5, respectively. In com parison to the reported B M T
dielectric constant - 25, the errors o f the estim ated dielectric constants are 8% , 10%, and
8%. U sing the simple log rule, such a result is considered to be quite satisfactory.
T o obtain the loss data in Table 5.1, data from the cavity perturbation m ethod were
m easured at 10 GHz. For the data from the post resonance technique, the m easurem ent
frequencies are different for different com positions because o f error in designing the
sam ple dim ensions. Since the loss value may change w ith the frequency, the loss data were
norm alized into 10 GHz values by assum ing tanS proportional to the frequency (W akino et
al., 1986). It is obvious that the Q factor data o f com posite sam ples in Table 5.1 are far
from the reasonable Q values, w hile the Q factors for both pure PE and B M T are higher
10
V
V
^
c
*
©
^
^
8
58.1% BMT
37.5% BMT
19.3% BMT
9.05% BMT
100% PE
o
A
o
A
+
Dielectric
Constant
<*><>0000. o
,°0°cd^oc^ cb^P3:b ^ 5 ^
A^ A
a AAa Aa ^
AAAAA/ia A M a AAa ^AAa M
a Aa a Aa Aa AAa ^Aa AAAAA
_ ^.+++++-H-+++"^'^^"+~*'+"*_*"++'H-+ I+-(^+'H-+++++++++4-H--H-H-+^
J
0
8
Figure 5.1.
[
I
L
I
«
1
i
»
I
I
1
I
I
I
I
I
10
11
Frequency (GHz)
I
I
I
12
I
I—
I—
L
13
Dielectric constants o f BMT-PE composites measured by the SI 1 and S21 technique.
O
□
10
10
Eq. (5.3)
Eq. (5.1)
Dielectric
Constant
+
S l l & S21 technique
Perturbation technique
Post resonance technique
\Eq. (5.2)
0
20
40
60
80
100
Volume Percentage of BMT (%)
Figure 5.2.
Comparison o f experimental data and theoretical expressions.
30.0
SI 1 and S21 Technique
Dielectric
Constant
25.0
23
20.0
15.0
Log rule
10.0
5.00
0.00
80
20
40
60
Volume Percentage of BMT (%)
Figure 5.3.
100
Estimation o f BMT dielectric constant. Data were measured by the
SI 1 and S21 technique.
£
O'
30.0
Cavity Perturbation Technique
Constant
25.0
22.5
20.0
Dielectric
15.0
Log rule
10.0
5.00
0.00
0
Figure 5.4.
60
80
40
20
Volume Percentage of BMT (%)
100
Estimation of BMT dielectric constant. Data were measured by the
cavity perturbation technique.
£
30.0
27
Post Resonance Technique
Constant
25.0
20.0
Dielectric
15.0
Log rule
10.0
5.00
0.00
60
80
100
Volume Percentage of BMT (%)
Figure 5.5.
Estimation o f BMT dielectric constant. Data were measured by the
post resonance technique.
149
than 4,000 at 10 GHz. A s m entioned before, the Q factors are deteriorated by the surface
condition o f the com posite samples. Therefore, the com posite m ethod is not adequate for
estim ating the quality factor o f pure material.
Figure 5.6 is the m easured loss data by the SI 1 and S21 technique. The loss tangent
goes to negative values because o f limitations on loss m easurem ent, as m entioned in
chapter 2.
5.5. Sum m ary
D ielectric properties o f composite m aterials with various volum e percentages o f B M T
and PE have been investigated in X band frequencies. It was found that the m easured
results on dielectric constants agree with the three mixture rules. W ith the estim ation o f the
dielectric constant o f 100% volume percentage B M T by the logarithm ic mixture, the
pow der m ixture method gave a reasonable prediction o f the dielectric constant o f BM T
ceram ic at X band frequencies. It was proved that the pow der m ixture method can not be
used to estim ate the Q factor o f pure material. T he accuracy o f dielectric constant
m easurem ent by the S 11 and S21 technique was confirmed.
0.3
- 5 8 . 1 % BMT
-- 37.5% BMT
- 19.3% BMT
9.05% BMT
- 100% PE
Loss Tangent
0.2
. oo
0.1
"O’ 'o .•
-
'O -
+.*
0.1
8
Figure 5.6.
9
11
10
Frequency (GHz)
12
13
Dielectric loss of BMT-PE composites measured by the SI 1 and S2I technique.
151
Chapter 6
C O N C L U SIO N S AN D FU T U R E W O RK
6.1. C onclusions
The m easurem ent techniques and the adequacy o f m easured materials for the
substrates o f HTSC films are concluded in the followings.
6.1.1. M easurem ent Techniques
A system atic com parison betw een different m icrow ave resonance techniques for
dielectric properties m easurem ent has been given in this thesis. The post resonance
technique is well developed and gives accurate m easurem ents on the dielectric constant, but
surface resistance o f m etal shields should be precisely m easured to ensure accurate
m easurem ent on loss tangent due to the high conductor loss. T he cylinder cavity resonance
technique can accurately m easure the loss tangent, but very complicated com puter w ork is
required for dielectric constant measurem ent and the accuracy is still unknow n. The
w aveguide reflection resonance method can also m easure the dielectric loss, but it has
higher m easurem ent error than the cylinder cavity m ethod, and dielectric constant
m easurem ent is not available. The cavity perturbation technique requires only a small piece
sample for dielectric properties measurement and gives accurate m easurem ent o f dielectric
constant, but its m easurem ent capability on loss tangent is limited. The accuracy o f on the
m easurem ent o f dielectric constant and dielectric loss by different microwave techniques
has been proved.
152
The sim ple Itoh and Rudokas M odel has been m odified to give correct prediction o f
the resonant frequency and reasonable computations on conductor and radiation losses o f
the parallel plate dielectric resonator. By applying the parallel plate resonance technique to
dielectric properties measurements, both dielectric constant and loss tangent can be
m easured. Unlike the cylinder cavity dielectric resonator where only com plicated field
expressions are available, the field distributions the parallel plate resonator can be roughly
described by the sim ple model and the high conductor loss problem o f the post resonance
technique is not exhibited.
6.1.2. Characteristics o f the Potential Substrate M aterials
The m icrow ave properties o f som e potential substrate materials o f the perovskite
oxide family and the perovskite fluoride family for the high Tc superconductor thin films
have been m easured at both room tem perature and liquid nitrogen tem perature. A
sum m ary on these substrate m aterials is listed as follows:
BMT: The B M T is the top choice for the application o f microwave antenna, which has
very high Q value and an adequate dielectric constant. The high melting tem perature of
B M T m akes its single crystal grow n by the Czochralski technique very difficult. However,
only a m m diam eter crystal is required for the antenna application and good quality m m
crystal fiber can be grow n by the LH PG technique.
SAT: T he SA T is the best candidate for the application requiring low dielectric
constant. It has very good lattice and thermal expansion m atches with the Y BCO
superconductor. Its low dielectric constant makes it a very attractive substrate material for
the applications o f high speed and m ultilayer design. The m ain disadvantage o f SAT may
be its higher m elting temperature (1,908 °C).
153
SAN: The SA N also has good lattice and therm al expansion m atches w ith the YBCO.
Its dielectric constant (18.2) is between that for the applications requiring low er and higher
dielectric constants.
SGT: The SG T has shown the potential for antenna application. Although its Q factor
o f ceram ic sam ple is less than 10,000, the Q factor of its single crystals is expected to be
higher than that o f the ceram ic samples and should be close to o r over 10,000. Its dielectric
constant also fits the applications requiring higher dielectric constants.
SA N -LA : The SA N -LA which has good lattice match with the superconductor has
also show n the potential for antenna application. It has a Q factor close to that o f SGT. In
addition, its m elting tem peratures is much low er than that o f the B M T for single crystal
growth. It can be used for higher dielectric constant applications, too.
SA N -N G ; The SA T -N G has the low est m elting tem perature (1,582 °C) am ong the
A (B iB 2 ) 0 3 fam ily for easy single crystal growth. It also has good m atches w ith the
Y BCO on lattice and therm al expansion. H ow ever, it also has the low est Q factor at liquid
nitrogen tem perature.
SA T-N G , SA T-LA , and SA N -SA T-N G : Their dielectric constants are m ore adequate
for the applications requiring higher dielectric constant values. A ll o f them have lattice
m ism atch w ith the Y B CO superconductor less than 3 % . They also have good therm al
expansion m atch w ith the YBCO, except the SA T-LA which has a large therm al expansion
m ism atch w ith the superconductor.
L M A i 1 O 1 9 Fam ily: The LM A, C G O , and N M G A are also good choices for low
dielectric constant devices. The LM A also has good lattice and therm al expansion matches
w ith the Y BCO . T he N M G A has higher lattice mismatch w ith the Y BCO than that o f SAT
and LM A. A lthough N M G A have higher dielectric constant (13.7) than that o f SAT, it is
still quite attractive for low dielectric constant devices by com paring with other higher
154
dielectric constant materials. The dielectric constant o f CGO is less than that o f SAT,
How ever, it has a higher lattice mism atch and very large thermal expansion m ismatch w ith
the Y BCO . The dielectric constant o f CGA is close to that of SAN. T he Q factors o f LM A ,
CG A, and C G O need to be further studied.
K M F 3 Family: The K M gF 3 also has a very high Q value for antenna application. It
has a better lattice match w ith the YBCO than that o f the BMT. In addition, both of
K M gF 3 and K ZnF 3 have shown very low dielectric constant for high speed and m ultilayer
devices. How ever, the K Z nF 3 has a higher lattice m ism atch with superconductor and both
K M gF 3 and K ZnF 3 have the problem s o f large therm al expansion m ism atch and o f
reacting w ith the Y BCO superconductor.
All the above m aterials have shown good potential as the substrate materials for
H TSC thin films.
It w as found that the variation o f the dielectric constant with frequency is very sm all
for the above materials and the loss tangent is roughly proportional to the frequency fo r the
perovskite oxide substrate m aterials with frequency around the order o f several gigahertz.
T he m icrowave dielectric properties o f the B M T -PE com posites have been studied
using three different pow der m ixture rules. It w as found that the experim ental results on
dielectric constants roughly agree with the logarithm ic mixture rule, M axw ell's m ixture
rule, and the newly reported m ixing equation by W akino et al.. The best fit was exhibited to
the logarithm ic mixture rule. B y extrapolating the dielectric constants o f various com posite
com pounds, the logarithmic mixture gave a reasonable prediction o f the dielectric constant
o f B M T ceram ic. The prediction o f Q value by the pow der m ixture m ethod was found not
adequate.
155
6.2. Future W ork
T he cylinder cavity resonance m ethod has the advantages o f no radiation loss, low
conductor loss, and low measurem ent error on Q factor m easurem ent, but the available
equations for field distributions are complicated. Great interest exists to simplify the
com plicated calculation for the cylinder cavity technique for dielectric constant
m easurem ent.
T he LM A, C G A , and CGO have show n good potential as the substrate of YBCO.
T he precise m easurem ent o f Q factor o f these materials is not reported in this thesis
because large enough sam ples were not available. Further study on their Q factors is
w orthw hile. This thesis focused on the m easurem ent o f ceram ic sam ples as the single
crystal samples are difficult to obtain. It would be w orthw hile to m easure the dielectric
properties o f single crystal samples, as low er dielectric losses than those o f ceram ic
sam ples should exist in crystal samples. In addition, the m easurem ents by cavity
perturbation technique have shown unreasonable higher dielectric constants on the single
crystal fibers o f som e materials than on ceram ic samples. The further study on the
dielectric constant values o f single crystals is necessary.
The LM A i 1 O 1 9 fam ily has show n the trend o f low dielectric constant values. Further
study on com pounds in this family to find som e new potential substrate m aterials for low
dielectric constant devices is worthwhile. The other fam ily which has shown low dielectric
constants and w orths further studied is the fluoride family.
Recently, Shannon (1993) has calculated the dielectric constants for about 100
com pounds by using an ion polarizability additivity rule. By studying, improving, and
revising the shannon's model, a list o f potential new substrate materials can be developed.
Such an approach can provide a w ider choice materials for the desired applications in the
future developm ent o f dielectric based devices.
156
F o r the m ultilayer design, the dielectric will be grow n as a form o f thin films. Since
the dielectric properties o f thin film m ay be different from that o f the b u lk crystal, in
addition to the m easurem ent techniques used for bulk sam ple, the developm ent o f a
nondestructive m easurem ent m ethod on the dielectric film will be required in the future. By
the w ay, the technique for study the m icrow ave properties o f superconductor film
(m icrow ave absorption and surface resistance) is going to be necessary and worths to be
paid attention to, too.
157
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VITA
Jyh Sheen was bom Septem ber 23, 1962, in H uW ei, Taiw an, Republic of China. He
received his prim ary and secondary education in ChungLi, Taiwan. H e received his
B achelor o f Science Degree in Electrophysics from the N ational Chiao T ung University,
H sinChu, Taiw an, in 1984. He entered the A ir Force o f Republic o f C hina in 1984. The
year after he retired from the A ir Force in 1986, he was a m athem atics teacher in Liu Ho
Technical H igh School, ChungLi, Taiw an. He received his M aster o f Science Degree in
Electrical Engineering from the N ew Jersey Institute o f Technology, N ew Jersey, in M ay
1989. Since then, he has been enrolled in the D epartm ent o f Electrical Engineering at The
Pennsylvania State University as a doctoral student.
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