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Ionic conductivity of alkali oxide glasses at microwave frequencies

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IONIC CONDUCTIVITY OF ALKALI OXIDE GLASSES AT
MICROWAVE FREQUENCIES
by
Sumithra Krishnaswami
A Dissertation
Presented to the Graduate and Research Committee
o f Lehigh University
in Candidacy for the degree o f
Doctor o f Philosophy
in
Materials Science and Engineering
Lehigh University
June 2001
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CERTIFICATE OF APPROVAL
A pproved and recom m ended for acceptance as a dissertation in partial fulfillm ent
o f the requirem ents for the degree o f D octor o f Philosophy.
3
£jtro
Date
LS
D issertation Advisor
Special com m ittee directing the doctoral research o f M s. Sum ithra Kxishnaswami:
A ccepted Date
Dr. Jk^fcriTTTChairper
airperson
Dr. S.G. Cargill
i-f . i l \
C_ l\ f u
J
Dr. H.M . Chan
Dr. D.M . Sm yth
,
h 1/
Uj&J.
Dr. C.M . W en"
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Dedicated to my parents
iii
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ACKNOWLEDGEMENTS
I would like to take this opportunity to thank m y advisor, Dr. H im anshu Jain, who
has been a source o f inspiration at every step o f my work. His advanced and profound
know ledge in glass science has been a constant source o f invaluable guidance, without
which this work would never have been com pleted. I also thank the National Science
Foundation for financial support o f this program .
I would like to extend m y sincere gratitude to the m em bers o f m y dissertation
com m ittee, Drs. Donald Sm yth, Helen C han, Slade Cargill and Claude W eil for their
adv ice throughout this research work. M y special thanks to Drs. Jim Baker-Jarvis and
John G rosvenor o f N IST, Boulder, CO for their help in setting up the microwave
m easurem ent system . The collaboration with Dr. E.I. Kam itsos for ER m easurem ents is
also greatly appreciated.
1 also thank Drs. O. Kanert, R. K euchler, A.S. Now ick and K. Ngai for their very
useful and stim ulating discussions on various issues o f this research w ork. M any people
have given me helpful suggestions and I w ould like to acknow ledge them here: Drs.
Richard Decker and M isha Shirokov for their help in m aking m e understand the complex
w orld o f m icrowave field distributions and interactions; Drs. R. V enkataram an and M.H.
W hite for their guidance w ith com plicated m athem atical analysis.
I am also grateful to all m em bers o f the group, both past and present, who all have
contributed in som e m anner: Dr. Charles Hsieh, Justin Jones, Dr. Jenny Peters, Faisal
iv
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Alamgir, H assan M oaw ad, Dr. Yasuo Ito, and G eorge Shannon. I thank them all for their
patience and m aking m y stay in W H339B a pleasurable one.
This will be incom plete w ithout thanking m y parents, Sulochana and Srinivasan
Kxishnaswami for their constant support, love and encouragem ent, without w hich 1 would
not have com e this far. I also thank my sister and brother for long phone conversations
over the w eekend. Last o f all I would like to say my special thanks to m y husband,
Vickram V athulya. who has been a pillar o f strength throughout m y work. His continuous
support, m otivation and love are deeply appreciated. Thank you for being there for me!
v
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TABLE OF CONTENTS
C E R T IF IC A T E O F A P P R O V A L
ii
ACKNOW LEDGEM ENT
iv
TABLE OF CONTENTS
vi
L IS T O F F IG U R E S
x
L IS T O F T A B L E S
xv
L IS T O F A C R O N Y M S
xvi
A BSTRA CT
I
C H A P T E R 1 IN T R O D U C T IO N
3
1.1 Introduction
3
1.2 O utline o f the dissertation
5
C H A P T E R 2 B A C K G R O U N D AND L IT E R A T U R E R E V IE W
2.1 The G lassy State
7
2.1.1 Glass Formation
7
2.2 G lass Structure
9
2.2.1 Binary Silicate G lasses
10
2.2.2 Binary G erm anate G lasses
11
2.3 O rigin o f Ionic Conduction in O xide Glasses
12
2.3.1 Introduction
12
2.3.2 DC Conductivity
16
2.3.3 Frequency-dependent Conductivity at Low Frequencies
20
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2.3.4 A D W P Model for NCL Low T em perature C onductivity
23
2.3.5 Ionic C onductivity at M icrow ave Frequency
25
C H A P T E R 3 O B JE C T IV E S AND O V E R A L L A P P R O A C H
39
3.1 O bjectives
39
3.2 O verall A pproach
39
C H A P T E R 4 E X P E R IM E N T A L T E C H N IQ U E S
4.1 G lass Preparation
42
42
4.1.1 Lithium Silicate G lass
42
4.1.2 .rK :0-( I -.r)GeO: Glass Series
43
4.2 A udio and RF Electrical M easurem ents
43
4.2.1 Principle
43
4.2.2 Experim ental set-up
44
4.3 M icrow ave Electrical M easurem ents
45
4.3.1 Electrom agnetic W ave Propagation
46
4.3.2 W ave Propagation in a T ransm ission Line
4S
4.3.3 Principle o f S-param eter M easurem ent
51
4.3.4 Experim ental set-up
53
4.4 Far Infrared M easurem ents
54
4.4.1 A nalysis o f IR R eflectance D ata
56
4.4.2 Experim ental set-up
58
C H A P T E R S A D W P M O D E L .AND ITS P R E D IC T IO N S
5.1 O rigin and Characteristics o f .ADWP M odel
vii
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6S
68
5.2 Predictions o f the A D W P M odel
5.2.1 ADW P Conductivity in 4K <T<150K
5.3 Influence o f Fitting Param eters
69
72
74
5.3.1 V ariation o f Vm
74
5.3.2 V ariation o f Am
75
5.3.3 V ariation o f t 0
75
5.3.4 Variation o fy
76
5.3.5 Variation o f V0
76
C H A PT E R 6 EX PER IM EN TA L RESULTS AN D D A TA A N A L Y SIS
6.1 Lithium Silicate Glass
S3
S3
6.1.1 Low Frequency Conductivity (H z-kH z)
S3
6.1.2 M icrowave C onductivity (50M H z-G H z)
S5
6.1.2.1 Data Analysis using A D W P M odel
6.1.3 Far Infrared C onductivity (>0.5TH z)
6.2 Potassium Germ anate Glass Series
6.2.1 Low Frequency Conductivity (H z-kH z)
S6
89
91
91
6.2 .1 .1 Scaling Properties o f C onductivity Spectra
92
6.2.1.2 Concept o f M ism atch and R elaxation
94
6.2.2 M icrowave C onductivity (100M H z-3G H z)
6.2.2.1
Data Analysis using A D W P M odel
97
99
C H A P T E R 7 FR EQ U EN C Y AN D T E M PE R A T U R E D E PE N D E N C E O F ADW P
127
C O N FIG U R A T IO N S
viii
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7 .1 Failure o f Classical Dielectric Loss Theories
7.1.1 Failure o f Scaling Laws
129
131
7 .1.2 A pplicability o f CM R M odel to Ionic C onductivity o f Ceram ics and
G lasses
133
7.2 The "Jellyfish’ M echanism at M icrow ave frequencies
135
7.3 C om position dependence o f M icrow ave C onductivity
140
7.4 C onnection to Far Infrared Conductivity
143
C H A P T E R 8 C O N C L U S IO N S
152
REFEREN CES
154
A p p en d ix A C O A X IA L T R A N S M IS S IO N A IR L IN E
166
A p p e n d ix B S T E P -B Y -S T E P IN S T R U C T IO N O F M IC R O W A V E
M E A S U R E M E N T U SIN G N E T W O R K A N A L Y Z E R H P 8753C
167
A p p en d ix C P U B L IC A T IO N S
169
V IT A
170
ix
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LIST OF FIGURES
Figure 2 .1: Specific volum e versus tem perature curves for glass form ation.
30
Figure 2.2: Structure o f (a) C rystalline, (b) G lassy A 2O 3 com pound.
31
Figure 2.3: (a) M odification o f silicate netw ork upon addition o f N a ions, (b) structure o f
sodium silicate glass.
32
Figure 2.4: Form ation o f 4-fold and 6-fold coordinated Ge atoms in binary germ anate
glasses due to the addition o f alkali oxide.
33
Figure 2.5: Conductivity, a and electric m odulus, M" as a function o f frequency for
lithium triborate glass.
34
Figure 2.6: Anderson-Stuart M odel o f ionic m igration in alkali silicate glasses.
35
Figure 2. 7: Haven Ratio, H r as a function o f Na^O concentration in sodium borate
glass.
36
Figure 2.S: C onductivity versus frequency plot o f sodium trisilicate glass at different
tem peratures.
37
Figure 2.9: C onductivity versus frequency plot o f sodium trisilicate glass show ing NCL
region betw een 4K. and 142K.
38
Figure 4.1: Schem atic representation o f W heatstone bridge w ith sam ple connected to one
arm o f the network.
60
Figure 4.2: Schem atics o f low frequency electrical configuration (a) three term inal
electrode on sample, (b) circuit connections.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
61
Figure 4.3: T ransm ission line for transverse electrom agnetic propagation (a) short piece
o f transm ission line, (b) lum ped-elem ent circuit.
62
Figure 4.4: Schem atic representation o f V ector N etw ork A nalyzer.
63
Figure 4.5: (a) T w o-port configuration, (b) flowgraph o f scattering param eters.
64
Figure 4.6: (a) Experim ental arrangem ent o f m icrow ave m easurem ent, (b) geom etry o f
sample and coaxial transm ission line.
65
Figure 4.7: C om parison betw een ICramers-Kronig and reflectivity m ethod for K.2O .2B2O 3
glass system in the frequency range 30-2000 c m '1.
66
Figure 4.8: (a) B ruker 113 v Fourier-transform vacuum spectrom eter for far infrared
reflectivity m easurem ents, (b) off-norm al sam ple arrangem ent.
67
Figure 5.1: Schem atics o f asym m etric double well potential. V and A are the barrier
height and asym m etry energy respectively.
77
Figure 5.2: V ariation o f ADW P conductivity with frequency for different Vm
values.
78
Figure 5.3: Tem perature dependence o f sim ulated .ADWP conductivity for different
asym m etry energy values, A>.
79
Figure 5.4: V ariation o f conductivity with (a) frequency and (b) tem perature for various
t 0 values.
80
Figure 5.5: D ependence o f AD W P conductivity on 7 at different frequencies.
81
Figure 5.6: D istribution function g(V)= 1/V0 sech(V /V 0) as a function o f barrier energy,
V for tw o different V0 = 50 K. and 150 K. values.
XI
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82
Figure 6.1: Low frequency ionic conductivity o f lithium silicate glass o f com position
35Li; 0 - 3 A l;0 3 - lP ;0 5-6 1 S i0 2 .
104
Figure 6.2: Tem perature dependence o f conductivity o f 35L i:0 -3A l203- lP :0 ;-61Si02
glass for frequencies betw een 100 Hz and 50 kHz.
105
Figure 6.3: Variation o f conductivity with tem perature in the region o f 4K < T < 100K.
for frequencies 50Hz, 100Hz, 500Hz, 1kHz, 5kHz, 10kHz and 20kHz.
106
Figure 6.4: M icrowave conductivity o f S S L iiO G A H O s-^ O sfo lS iC h glass from RT to
400K.
Figure
107
6.5:
Tem perature
dependence
of
m icrow ave
conductivity,
ctMw
for
35Li:0 - 3 A l:0 : l P : 0 5-6 lS i0 ; at frequencies 100 MHz, 500 M Hz. 1 GHz. 2 GHz and 3
GHz.
108
Figure 6.6 : Frequency dependence o f em pirically determ ined constants, (a) a and
(b )B '.
109
Figure 6.": (a) Com parison o f em pirically extrapolated low f - low T conductivity data at
RT and experim ental high f - high T data, (b) Extrapolation o f low f - low T using the
form alism o f .ADWP.
110
Figure 6 .8: (a) Com puter sim ulation o f A D W P conductivity using Eqn. (5.7) as a
function o f tem perature for several frequencies, (b) Frequency dependence o f
ctadwp
300K.
at
111
Figure 6.9: (a) Variation o f sim ulated <j awdp w ith tem perature for frequencies indicated
using Eqn. (5.7). (b) Predicted frequency dependence o f c t a d w p at 300K.
Figure 6.10: F ar infrared conductivity data o f 35L i20-3A l203-lP20s-61Si02 glass
xii
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112
a tR T .
113
Figure 6.11: C onductivity as a function o f frequency for lithium silicate glass.
114
Figure 6.12: Frequency dependence o f a for 0 .2 4 7 K i0 0 .7 5 3 G e 0 2 glass for tem peratures
from 1 0 0 K to 4 0 0 K .
115
Figure 6.13: ‘M aster’ plot o f norm alized conductivity,
g / ctjc
versus
xq)/T ctjc
for three
di fferent com positions o f K iO -G eO : glass.
116
Figure 6.14: M aster plots o f (a) 1% and (b) 1 l% G d 3^ doped C eO : crystal.
117
Figure 6.15: Scaling plots for 0.247K : O - 0.753G eO : glass at tem peratures 394.7K. and
346.3K .
118
Figure 6.16: Frequency dependence o f m icrowave conductivity o f 0.247K ;O -0.753G eO :
glass betw een 300K. to 400K..
119
Figure 6.17: Frequency dependence o f m icrow ave conductivity o f 0.074K :0-0.926G eO :
glass betw een 300K to 400K.
120
Figure 6.18: C om parison o f em pirically extrapolated low f - low T data and experim ental
high f - high T data for 24.7 m ol% and 7.4 m ol% K ;0 glass.
Figure
6.19:
Tem perature
dependence
of
m icrow ave
121
conductivity
of
0.247KiO-0.753GeO 2 glass com position at 5x10' Hz, 1x 10s H z, 5x10s Hz, lx lO 9 Hz,
2 x l0 9 Hz and 3 x l0 9 Hz.
122
Figure 6.20: D ependence o f tem perature exponent ‘y’.
123
Figure 6.21: C om position dependence o f m icrow ave conductivity for the K^O-GeO; glass
series at 100 M Hz, 512 M Hz, and 1 G H z at three different tem peratures.
X III
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124
Figure 6.22: Alkali concentration dependence o f A D W P concentration in lithium
germ anate glass.
125
Figure 6.23: V ariation o f ADW P concentration w ith K ? 0 content in the glass at
m icrow ave frequencies.
126
Figure 7 .1: Real part o f conductivity as a function o f frequency o f a lithium silicate
glass.
145
Figure 7.2: M aster plot for GdJ^ - CeO : at three doping levels.
146
Figure 7.3: Com parison betw een experim ental m icrow ave conductivity and .ADWP
model for 24.7K :0 -7 5 .3 G e0 :.
147
Figure '.4 : Influence o f L D R conductivity at m icrow ave frequencies.
148
Figure 7.5: C om parison o f com position dependence o f Ode,
ct.mw
and
gltlf
for K-Ge glass
series.
149
Figure 7.6: Schem atics o f jellyfish structure at m icrow ave frequencies.
150
Figure 7.7: Contribution o f AD W P conductivity to high frequency conductivity in
3 5 Li2 0 -3A l;0 3 - l P ; 0 5 -6 lS i0 2 glass at RT.
151
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LIST OF TABLES
Table 6.1: C om parison ot' tem perature exponent ‘y’ for K -G e glass series at low T - low f
and high T - high f regions.
101
Table 6.2: Sum m ary o f AD W P fitting param eters at m icrow ave frequencies for K-Ge
glass series and lithium silicate glass.
102
Table 6.3: Values o f A D W P concentration, N for K-Ge glass series.
103
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LIST OF ACRONYMS
ADW P
Asym m etric Double W ell Potential
AD W PC
Asym m etric Double Well Potential Configuration
BO
Bridging oxygen
CM IC
Cyrstalline M icrowave Integrated Chip
CM R
Concept o f M ismatch and Relaxation
CN
C oordination Num ber
DCR
D iffusion-controlled Relaxation
DL’T
Device under test
ECR
Electrical Conductivity Relaxation
FIR
Far Infrared
GM IC
Glass M icrow ave Integrated Chip
JRM
Jum p Relaxation Model
K-K
K arm ers-K ronig
LO
Longitudinal Optical
MW
M icrowave
NBO
N on-bridging oxygen
XCL
N early Constant Loss
RT
Room tem perature
TEM
T ransverse Electromagnetic
TLS
Two Level System
TO
T ransverse Optical
UDR
Universal Dynamic Response
VNA
V ector N etwork Analyzer
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A B ST R A C T
The use o f glass as additive in m ultilayered dielectric structures, as substrate in
m icrow ave integrated chips for com m unication devices etc. has all been a m otivation to
study the m icrow ave dielectric properties o f glass. Early studies o f high frequency
conductivity, particularly in the m icrow ave region conducted at one or tw o frequencies
and at room tem perature w ere suggested to be due to five independent sources o f
dielectric loss m echanism s based on m igration, vibration, and/or deform ation associated
with ion m ovem ent in glasses and glass-ceram ics. However, these theories are either
inconsistent with experim ental data or lack satisfactory m icroscopic description.
The m icrow ave conductivity, avtw. o f oxide glasses is show n to increase linearly
with frequency but with w eak therm al activation. This is sim ilar to the low tem perature
low frequency conductivity, where the m echanism o f conduction is established to
originate from jellvfish-like localized fluctuations o f atoms in asym m etric double well
potential (ADW P) configurations. In other words, the high frequency - high tem perature
dielectric loss in oxide glasses can be understood in terms o f the jelly fish conduction.
C om puter sim ulations o f the A D W P form ulation explain the observed linear frequency
dependence and pow er law tem perature dependence o f m icrow ave conductivity well.
Thus the conduction m echanism at m icrow ave frequencies can be described very well
by the localized fluctuations o f jellyfish structures. Careful exam ination o f the AD W P
param eters especially the pre-factor to relaxation tim e, the asym m etry energy, and the
concentration o f A D W P structures in the two regions, viz. low f - low T and high f -
1
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high T. reveal that different kinds o f jellyfish structures are responsible for conduction in
the tw o regions. From the com position dependence o f ctmw and A D W P concentration,
w e conclude that not all alkali ions participate in conduction, but only a fraction that are
co-ordinated to the netw ork in som e special configuration contribute to the jellyfish
m echanism at M W frequencies. The effective region o f the jelly fish , which consists o f
the m odifier as well as the netw ork atom s, is estim ated to be approxim ately -2 5 A or
less.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CHAPTER 1 INTRODUCTION
1.1 IN T R O D U C TIO N
The transport properties o f oxide glasses have been o f m uch interest for several
decades because o f their application in glass technology. Recent developm ents include
the em ergence o f low energy loss devices [1.2], electrical insulators [3,4], nuclear waste
forms [5.6], fast ion conductors for fuel cells and batteries [7,8,9], solid electrolytes
[10.11.12.13.14] etc.
A nother area o f considerable interest in using glass and glass
ceram ics with low conductivity is in high-speed com puters and com m unication systems
operating at m icrowave frequencies. The replacem ent o f crystalline m icrow ave integrated
chip (CM IC ) by glass m icrowave integrated chip (GM IC) is m otivated by the attractive
features like low loss and conductivity o f glass at these frequencies. On the other hand,
highly conducting glasses have found applications in m icrow ave sintering, firing and
processing o f ceram ics [15,16]. .Among all these efforts, the use o f m icrow ave radiation
for sintering o f ceram ics has drawn m ost attention. However, there exist som e unsolved
problem s
like the catastrophical phenom enon o f thermal runaw ay hindering the
developm ent o f this em erging technology. V ery recently, m icrow ave processing o f
glasses is also being established [17,18], but it is unclear how the above phenom enon will
affect this technology.
Ion dynam ics describes the m ovem ent o f ions in a solid. The long range m igration
o f ions is characterized bv: radiotracer diffusion D* and dc electrical conductivity a , o f
w hich the latter is m ore popular because o f the ease o f the experim ent.
U nlike ionic
3
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
conduction in crystalline solids for which the theoretical and experim ental w ork has been
well established [for review s, see 19,20,21], the m echanism o f ion transport in glasses is
still far from clear. This is m ainly due to the fact that glasses do not possess well defined
or ordered structure as crystals do and that their disordered structure is not well
understood yet.
Ionic transport in crystals like N'aCl can be easily understood through
therm ally activated defect pairs (Schottky type or Frenkel type). The activation energy
for conduction is, therefore com posed o f defect-pair form ation and defect m igration
energy, both o f which are often constant over a wide range o f com position because o f the
persistence o f crystal structure.
In contrast, the structure o f glass changes with the
addition o f doping ions, m aking it impossible to distinguish the form ation and m igration
energies. Also the lack o f long-range order in glasses m akes it difficult to investigate the
structure using conventional techniques like X -ray or neutron diffraction. As a result, a
clear picture o f ion m ovem ent in glasses is still lacking.
The com plete description o f ion dynam ics spans ion m ovem ents at short times
and/or lim ited distance. This inform ation can be characterized by observing the ionic
conductivity, a , over a w ide range o f angular frequency to (from dc to IR frequencies)
and tem perature T (cryogenic to the glass transition tem perature, T g). There is a large
volum e o f literature on the conductivity trends over lim ited segm ents o f co and T
spectrum on several crystals and oxide glasses [22,23,24,25,26,27], However, a
com prehensive understanding o f ion conduction in glasses is still under debate due to the
lack o f experim ental data over certain ranges o f frequency and tem perature. As a result,
theoretical m odels over the full range do not exist or often lack predictability. The
4
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
purpose o f this particular w ork is to bridge the gap betw een the existing regions o f
conductivity data at m icrow ave frequencies and to propose an ion conduction m odel for
this region.
1.2 O U T L IN E OF TH E DISSER TA T IO N
The organization o f this dissertation is as follows: A fter a b rie f introduction in
C hapter 1, a review o f general background and literature is presented in C hapter 2. It
includes the glass form ation principles, origin o f ionic conduction in oxide glasses,
different theories and m odels o f ion conduction in highly defective crystals and glasses at
different tem perature and frequency regim es. A detailed discussion on several existing
theories and m odels o f m icrow ave conduction is given in this chapter thereby leading to
the crucial questions that this dissertation intends to resolve.
C hapter 3 gives the
objectives o f the present work and the general approach o f the dissertation. The
experim ental techniques in this w ork are described in detail in C hapter 4. They include
glass preparation m ethod, and audio, radio, m icrow ave and far infrared frequency
m easurem ents. The principle and theory o f m icrow ave interactions w ith the am orphous
solid
are
discussed
extensively and a detailed
description
o f the experim ental
arrangem ent built in our laboratory is also presented.
In C hapter 5, the salient features o f the A sym m etric D ouble W ell Potential
(.ADWP) m odel along w ith its m athem atical predictions are discussed. The dependence
o f conductivity predicted by the m odel on various fitting param eters is also discussed
briefly. In C hapter 6, the electrical conductivity results at different frequencies and
5
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tem perature are presented for each o f the glass system s investigated, w hich include a
lithium silicate and a series o f potassium germ anate glasses. C hapter 7 forms the
discussion part o f the dissertation for the glass system s studied. Final conclusions are
given in C hapter 8. follow ed by the publications o f this research in the appendix.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CHAPTER 2
BACKGROUND AND LITERATURE REVIEW
2.1 TH E G L A SSY STATE
Scientific research has, for a long tim e, paid insufficient attention to solids o f non­
periodic structure. Although glass is one o f the earliest m aterials used, it is only in recent
decades that progress has been made tow ards developing theories o f the am orphous and
vitreous state. An am orphous substance, even w ith identical chem ical com position,
differs from its crystalline counterpart by its higher energy content. Unlike crystals,
where the atom s are arranged periodically exhibiting long-range order, am orphous solids
do not exhibit long-range order. In other w ords, the atom ic positions are strongly
disordered in an am orphous solid. A glass is an am orphous or a non-crystalline solid that
undergoes a glass transition phenomenon. A m ore general definition o f glass is given by
W ong and Angell [28]: “ Glass is an x-ray am orphous m aterial which exhibits the glass
transition, this being defined as that phenom enon in w hich a solid am orphous state
exhibits with changing tem perature a m ore or less sudden change in the derivative
therm odynam ic properties, such as heat capacity and expansion coefficient, from crystal­
like to liquid-like values.”
2.1.1 Glass Form ation
G lasses are usually formed by solidification from the m elt. The structure o f the
glasses can be clearly distinguished from that o f liquids, since glass structure is
effectively independent o f temperature. This can b e best seen by the plot o f the specific
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volum e o f the crystal, liquid and glass as a function o f tem perature as show n in fig. 2. 1.
On cooling the liquid, there is a discontinuous change in volum e at the m elting point if
the liquid crystallizes.
However, if no crystallization occurs, the volum e o f the liquid
decreases at about the same rate as above the m elting point until there is a decrease in the
expansion coefficient at a range o f tem perature called the glass transform ation range.
Below this tem perature range, the glass structure does not relax at the cooling rate. The
expansion coefficient for the giassv state is usually about the sam e as that o f the
crystalline solid. If slow er cooling rates are used so that the tim e available for the
structure to relax is increased, then the supercooled liquid persists to a lower tem perature
and a higher density glass results.
A concept useful in describing the properties o f the glass is the glass transition
tem perature. T... which corresponds to the tem perature at the intersection o f the curves o f
the supercooled
liquid and the glassy state in fig. 2.1. D ifferent cooling rates,
corresponding to different relaxation tim es, give rise to a different configuration in the
glassy state. In the transition range, the tim e for structural rearrangem ents is sim ilar in
m agnitude to that o f experim ental observations. As a result, the configuration o f glass in
this range changes slowly w ith tim e towards the equilibrium structure. At higher
tem peratures, the structure corresponding to equilibrium at any tem perature is achieved
rapidly. At substantially low er tem peratures, the structure o f the glass rem ains stable over
long periods o f time.
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2.2 G LA SS ST R U C TU R E
The essential aspect that differentiates an am orphous m aterial from a crystalline
solid
lies
in their structure. A
fundam ental difference
is evident from
atom ic
arrangem ents in the two solids. In 1932, Z achariasen [29] argued that since the
m echanical properties o f glasses are sim ilar to those o f the corresponding crystals, the
atom ic forces in both must be o f the same order. The diffuseness o f the x-ray diffraction
spectra o f glasses clearly indicated that glass itself has an infinitely large unit cell. Glass
would therefore consist o f a three-dim ensional random netw ork. A direct consequence o f
this random ness w ould result in glass having higher internal energy than the crystal.
Zachariasen believed that this difference in the internal energy should be small, since
otherw ise, there w ould be sufficient driving force tow ards crystallization. This smail
energy difference required an open and flexible structure.
Based on his views, he proposed a random netw ork m odel for an oxide glass,
which is w idely accepted. According to this m odel, a glass is view ed as a threedim ensional netw ork or array, lacking sym m etry and periodicity, in w hich no unit o f the
structure is repeated at regular intervals. In the case o f an oxide glass, the basic structural
unit is the oxygen polyhedra o f the netw ork form ing atom s like Si, Ge, B etc. A random
netw ork is then constructed by a set o f rules:
1) Each oxygen ion should be linked to not m ore than two cations.
2) The coordination num ber o f oxygen ions about the central cation m ust be small, 4
or less.
3) O xygen polyhedra share com ers, not edges o r faces.
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4) At least three com ers o f each polyhedron should be shared.
For instance, oxides o f the form ula ACF, A2O 3, A:Os etc. satisfy these rules and
can therefore form glass. The atom ic structural representation o f A^CT. com pound in the
crystalline and glassy forms is shown in fig. 2.2. Note that both the crystalline and the
glassy forms are com posed o f A O 3 triangles jo in ed to each other at com ers, except that
the glassy form has disorder introduced by changes in the A -O-A angles, which are called
bond angles, and slight changes in the A -0 bond length.
2.2.1 Binary Silicate Glasses
In a single-com ponent glass system like silica, all oxygen atom s are bridging
atom s (BO), i.e., they connect two polyhedral units. The addition o f alkali oxide to the
single com ponent glass results in the m odification o f the netw ork in that the alkalis enter
the glass as singly charged cations. The unit positive charge is satisfied by the form ation
o f an ionic bond with an oxygen atom , which is accom plished by breaking the bridge as
show n in fig. 2.3a. Thus each alkali is expected to create one nonbridging oxygen (NBO).
N ote that som e oxygens (BO) are connected to two silicon atom s w hile som e oxygens
(N BO ) are connected to one. The sodium atom s are situated in the voids o f the tetrahedra
in the vicinity o f the NBO for charge com pensation. The creation o f NBO in the netw ork
reduces connectivity and hence m ass-transport related properties such as diffusion,
electrical
conductivity,
chem ical
corrosion
etc
increase.
A
tw o-dim ensional
representation o f the structure o f sodium silicate glass is show n in fig. 2.3b.
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2.2.2 Binary G erm anate Glasses
The present interest in binary glass-form ing germ anates stem s
from the
anom alous changes in the properties when alkali oxides are introduced in the glass
netw ork. The appearance o f the extrem a in the properties as a function o f alkali oxide
content was first independently observed by Evstropev and Ivanov [30,31], Riebling [32]
and M urthy et al. [33,34], These groups studied various properties as a function o f
com position, such as density, viscosity, electrical conductivity, refractive index, chemical
stability and oxygen content. The observation o f either a m inim um or a maximum in
properties with com position was termed the ‘germ anate an o m aly '. It was proposed that
the addition o f alkali oxide (R ;0 ) to G eO : causes the partial conversion o f germ anium oxygen tetrahedra into octahedral units when the alkali content is upto 20 mol% as shown
in fig. 2.4. Higher concentrations o f R :0 cause the form ation o f nonbridging oxygens
(N BO ) containing germ anate tetrahedra [35.36]. Using x-rav diffraction, Sakka and
K am iya [37] established that for the system o f glasses o f the type R ;0 - G eO : (e.g., LHOG eO ;, N a :0 -G e O :, K iO -G eO : and R b :0 -G eO :), germ anium is in a fourfold coordinated
configuration for very low concentrations o f alkali ions (<2 m ol% ). As the concentration
o f alkali ion is increased, the fourfold germ anium (G eQ i) is converted to a sixfold
coordinated germ anium (GeOb) until about 20 m ol% o f the alkali ions. The am ount o f
sixfold germ anium is calculated by the relation [GeOs]= x /l-x . w here x is the alkali ion
concentration. The m axim um concentration o f the G eO b groups was calculated to be
around 25-30% for about 20 m ol% o f alkali oxide content.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
H ow ever, recent studies using x-ray photoelectron spectroscopy (XPS) show that
NBOs are form ed even w ith the addition o f very low alkali concentration ( - 2 m ol% )
along with the form ation o f GeOe units [38]. At concentrations >25 m ol% , the form ation
o f \ rBO atom s in Qn (n=bridging oxygens, 4-n=non-bridging oxygens) tetrahedral species
is the m ain m odification o f the structure, leading to progressive depolym erization o f the
network.
The coordination num ber o f germ anium around oxygen has been also studied
using other spectroscopic techniques [37,39,40]. It is very clear that the coordination
num ber (CN) o f Ge atom s changes with the addition o f the alkali ion. Infrared analysis o f
K ;0 -G eO ; glass system shows that the average C N o f Ge attains a m aximum o f s4 .3 2 at
-2 5 m ol% K ;0 . This is in good agreem ent w ith published results o f Sakka and Kam iya
[37] and Huang et al. [40] on the same glass system em ploying techniques like x-ray
diffraction and EXAFS.
2.3 O RIG IN O F IONIC C O N D U C T IO N IN O X ID E G LA SSES
2.3.1 Introduction
Ion dynam ics in glasses have been characterized by studying the diffusion o f
tracer ions (D ) or electrical conductivity (a ), though m ost workers prefer the latter
because o f the ease o f the experim ent [41]. T he frequency (f) and tem perature (T)
dependence o f electrical conductivity (cr) o f oxide glasses is im portant not only in their ac
applications but to gain an insight in the various conductivity m echanism s as w ell. For an
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ideal ionic conductor, the conductivity should be independent o f frequency and therm ally
activated, except at very high frequencies (~ 1012 Hz), w here it m ay increase due to ionic
vibrations as - o r , co(=27tf) being the angular frequency and becom e tem perature
independent. H ow ever, for m ost oxide glasses, this is not true and a com bination of
different pow er laws is obtained over the wide frequency range and tem perature range.
The phenom enon o f Electrical Conductivity R elaxation (ECR ) indicates the timedependent response o f charge m ovem ent to the application o f an external electric field.
M onovalent cations like alkali or silver ions are the most com m on m obile charge carriers
and are predom inantly responsible for charge transport, although other kinds o f mobile
charges m ay have a m inor contribution [42], It has been observed that ECR typically
occurs above audio frequency range depending on the m aterial and tem perature. This
observation is not surprising since at very low frequencies, the cations are practically
im m obile with respect to the applied electric field; hence very low or negligible electrical
relaxation.
Thus, electrical conductivity at high tem peratures (> room tem perature) was
em pirically described by Jonscher as made o f two com ponents [43]: (a) a dc plateau at
low frequencies w here a
is independent o f frequency and, (b) an ac region at
interm ediate frequencies w here a show s a pow er law dependence w ith frequency. This
behavior is observed in m any crystals and glasses and has com e to be know n as the
‘Universal Dynam ic R esponse' (U D R ). The overall frequency dependence o f a in this
lim ited frequency range can b e w ritten as:
a(co,T) = a (0 ) + Aco1
(2.1)
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
w here
cr(O)
is
the
frequency independent
dc
conductivity,
A
and
0 < s< l,
are
experim entally determ ined parameters. For m ost glasses and ionic crystals, s=0.5-0.6 [44]
and the observation o f transition from dc to ac conductivity is a function o f tem perature.
A careful analysis o f experimental data show s that Eqn. (2.1) is generally obeyed except
in the vicinity o f the transition [45,46]. T he sam e region is described som etim es as a
departure in the peak o f the imaginary part o f dielectric m odulus, M" (M * = l/e \ e* being
the com plex dielectric perm ittivity), versus frequency plot from the standard D ebye shape
[47.48]. M" is defined as:
coC ,G
M = — ;----- ^
r
G ' ~ (a C )~
(2.2)
w here. G. C and CQ are the conductance, capacitance and the vacuum equivalent
capacitance o f the sam ple. In general, the M" peak occurs at a frequency in the dc-»ac
transition region, as shown for lithium triborate glass at 132.1°C in fig. 2.5. The peak
occurs at cotdc~ l. w here rdL. (=C/G) is the M axwell relaxation time. The non-Debye
character o f M" peak is described in term s o f fCohlraush-W illiam s-W atts (KW W )
distribution corresponding to the electric field decay function in the tim e dom ain, given
by:
<p(t) = e x p [ - ( t / t KWWJ15]
(2.3)
w here 0< p < l is called the Kohlraush param eter and tkww is the characteristic relaxation
tim e. M oynihan et. al [47] showed that the above eqn. could be used to describe the ionic
conductivity relaxation in glass. The value o f P for a given m aterial can be determ ined
from the shape o f the M " peak. It has been show n that p is m obile ion concentration
14
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
dependent, and approaches a value o f 1.0 at low concentrations o f alkali ions [49,50], A
value o f [3<l.O reflects a deviation from the ideal behavior in the vicinity o f idC. In
addition to a and M ", there exist other forms o f representations o f electrical relaxation
such as com plex impedance, complex adm ittance, com plex susceptibility etc., all o f
which contain the sam e inform ation, but with som ew hat different em phasis. However, for
the sake o f consistency, we will use the electrical conductivity form alism throughout this
dissertation.
Interestingly, Now ick et. al observed in som e highly defective crystalline solids,
that when tem perature was lowered to cryogenic values (4K <T<200K ), the dc plateau
m oves out o f the frequency window and the observed low frequency conductivity
increases nearly linearly with cu. This behavior was observed by others in a wide variety
o f crystals and glasses, and has been quite extensively studied [51,52,53]. This region is
cailed the 'nearly constant loss’ region or NCL and has com e to be described as a second
universality. As a consequence, an alternate em pirical equation for describing a at low
frequencies was suggested, and can be written as:
ct(co, T) = <t0 + Aco4 ■+■A'co'
(2.4)
where, A' and s '= l are the em pirically determ ined constants.
Since the nature o f ion transport m echanism is specific to the region o f frequency
and tem perature, it is im portant to discuss the different theories and m odels o f conduction
that has been established at different regions.
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2.3.2
DC C onductivity
T he dc conductivity o f a crystal w ith one kind o f m obile ion is given by:
ct(0)
= (Z e )n .u
(2.5)
w here Ze is the charge, n, is the concentration and u is the m obility o f the m obile ion. If
conduction occurs via a defect m echanism such that at any given instant, only a fraction
(n,) o f the total ions (N) are m obile, then we can write
G
S
H
n = \'e x p < - —i-) = N e x p ( - M e x p ( - - ^ )
kT
kT
kT
( 2.6 )
w here G t- is the Gibbs free energy for the form ation o f a defect, S t- and H,- represent the
entropy and enthalpy o f formation respectively, kT has the usual m eaning. The m obility
o f the ion. u is related to the ion diffusivity (D) via the N em st-E instein relation expressed
as [ 5 4 ] :
u = — D = — fd : v m
kT
kT
(2.7)
Here f is the correlation factor, d is the jum p distance and vm is the jum p frequency. If v0
is the attem pt frequency and G m is the free energy o f m igration,
v ™ = v o exP(“ ^ f ) = exP(£ p ) exp(" - ^ r )
<2-8)
C om bining Eqs. (2.6-2.8) gives
g(Q) = N(k T > ad2Va 6XP(Sf
)CXP(~ H ' k T ^ m }
(2 ‘9)
If EoC=H,—Hm and the pre-exponential term is represented by a constant term. A, then the
dc conductivity can be expressed as:
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
where E^- is the dc activation energy. Thus one can see that dc conduction is strongly
therm ally activated in solids. Based on the structure o f glass, several m odels have been
proposed for explaining the long-range transport in glasses, some o f w hich are discussed
briefly.
The Stroma Electrolyte M odel
T h e strong electrolyte m odel assum es that all m obile ions participate in the
c o n d u c t i o n process and their concentration in Eqn. (2.5) is a constant independent o f
t e m p e r a t u r e . As a consequence, only the m obility o f the ions is responsible for EdCA n d e r s o n a n d Stuart (A-S) calculated from first pnnciples the activation energy for
c a t i o n i c m igration in alkali silicate glasses [55], A ccording to their m odel, the energy o f
m i g r a t i o n c o u l d be expressed principally as a sum o f two energy term s as show n by Eqn.
(2 .1 !). Ep. t h e electrostatic binding energy that must be first overcom e, and Es, the strain
energy r e q u i r e d to open up ‘doorw ays' in the structure large enough for the ions to pass
through. Therefore.
EA = E b + E s
(2. 11)
A convenient visualization o f this process by M artin and Angell [56] is show n in fig. 2.6.
In effect, EA is the difference betw een the bottom o f the energy w ell (w here the cation
norm ally resides) and the m axim um in energy where the cation is poised h a lf way
betw een neighboring sites.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Even though the A-S theory is w idely applied to Li" conducting glasses [57],
m ixed anion conductors [58] etc., it m erely depends on som e poorly defined or em pirical
param eters and does not take into account the details o f glass structure and jum p
m echanism [59].
The W eak Electrolyte (W -E) Model
Contrary to the A-S theory o f strong electrolytes, this m odel proposed by Ravaine
and Souquet [60] assum es that the ion m obility is independent o f glass com position and
therefore has only a small dependence on Ejc. They observed that large increases in the
conductivity in N a ;0 -S iO ; glasses are paralleled by large increases in the Na^O activity.
As a result, the dc conductivity and activation energy are strongly dependent on the
m obile ion concentration,
a Jca [ N a ‘ ]
(2. 12)
O ther support to the W-E model cam e from conductivity m easurem ents on
different glass system s with high alkali concentrations such as AgPCh-AgI [61], where
conductivity was found to depend greatly on the A g ion concentration. However, this
m odel could not explain conduction in oxide glasses w ith low concentration o f alkali
ions, since the addition o f alkali oxides to such glasses can cause drastic changes in the
glass structure and m obility.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
The D efect Model
Ionic transport in crystals usually involves the m igration o f defects like vacancies
or interstitials. For exam ple, these defects could be Li* vacancies in Lil, O2' vacancies in
CaO stabilized ZrO ;, or Ag* interstitials in AgBr. The interstitial cations m ay hop
directly betw een interstitial sites or in a pair-w ise m echanism displacing a cation from the
neighboring lattice site.
Haven and Verkerk [62] compared tracer diffusion coefficients, D*. w ith values
derived from the Nem st-Einstein equation
(2.13)
ne‘
They argued that when ionic m otion involves either a vacancy or an interstitialcy, the
value o f D will be influenced by correlation effects, w hereas Da will not. Corrections to
the N em st-Einstein equation were expressed by the Haven ratio,
Hr = f -
(2.14)
For typical sodium borate glasses, the value o f Hr lie around 0.3-0.4 for about 30
m ol% o f N a 20 [63] but approaches unity in nearly pure borate glass as show n in fig. 2.7.
The near unity value for Hr at low alkali content supports the idea that the alkalis m ove
m uch like interstitials. The distance betw een the alkali ions is large and hence their
m otions are random and not correlated. As the concentration o f alkali increases, their
m otion becom es highly correlated resulting in a decrease o f Hr. C learly the sim ple
interstitial m echanism is not appropriate even for alkali concentration o f ~5 m ol% . Early
discussion o f diffusive m ovem ents in glass assum ed various crystalline lattices to
19
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
represent glass structure and to calculate Hr for conduction m echanism . For example,
assum ing the num ber o f nearest neighbors to be six (hexagonal arrangem ent), the Hr was
found to be 0.67 for vacancy m echanism and 0.33 for interstitial m echanism [62].
Consequently, for concentrations >5 m ol% o f Na^O in fig. 2.7, a com bination o f both
vacancy and interstitial m echanism must be operating in such glasses. Subsequently Jain
et al. [64] m odified the idea o f the m ixed vacancy and interstitial m echanism by taking
into account the interactions betw een the m obile ions and their charge compensating
centers
in the
netw ork with
increasing concentration.
T his produced additional
correlation effects, w hich were able to explain the observed tem perature and composition
dependence o f Hr.
2.3.3 F requency-D ependent Conductivity at L ow Frequency - T h e UD R Region
C om m on trends in the frequency dependence o f ionic conductivity ct(co.T) o f
som e crystalline and oxide glasses have been reported. A typical exam ple o f the
absorptivity ' a ' (proportional to a ) variation as a function o f both to and T is shown for
N a iO o S iO : glass by W ong and Angell [65] in fig. 2.8. As can be seen, at low
frequencies, the cr(co) at room tem perature shows a flat plateau independent o f frequency,
which is described as dc conductivity a(0). The variation o f ct(0) at higher tem peratures
is well understood. The dc conductivity follows an A rrhenius plot in m ost oxide glasses
and som etim es a non-A rrhenius variation at high T in som e fast-ion conductors (FICs)
[66]. The variation o f a (0 ) with respect to the ion concentration ‘x ’ is still unclear. Earlier
20
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
w ork show s that there is a pow er-law increase o f EdC w ith x in som e binary oxide silicate
glasses, but exceptions have been reported for som e germ anate and other glasses [67,68],
As co increases, a increases follow ing a pow er law given by ctldr~Aco5 (i.e.
a ( c o ) » a 0 in Eqn. (2.1)) w ith s~0.5-0.6 and A being therm ally activated w ith energy
E<El]i;. N um erous theories have been proposed to explain this pow er law frequency
dependence o f ionic conductivity in glasses, and they fall into two m ain categories. The
first idea is based on a distribution o f relaxation tim es or potential barriers [69,70], while
the second is based on the non-random hopping o f ions resulting from ion-ion
interactions [71.72,73,74,75]. Since the first idea usually requires wide and unreasonable
range o f relaxation tim e distributions, at present it seem s that the interaction m odels are
m ore appealing though m uch debate is still going on.
Jum p Relaxation Model (JRM )
The jum p relaxation model proposed by Funke is based upon the existence o f
correlated forward and backw ard ion hops due to the retarded m ovem ents o f the
neighboring ions [71]. A fter an ion has jum ped from a site A to a vacant site B, two
com peting relaxation processes occur. T he central ion m ay hop back to A resulting in a
backw ard or an unsuccessful hop, or the hop m ay be successful and the surroundings
relax w ith respect to the newly occupied site B. The non-D ebye character o f the
relaxation is due to the fact that the tim e constant o f the back-hop process increases as
tim e progresses.
This assum ption leads to a frequency dependence o f conductivity as
21
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
expressed in Eqn. (2.1) in the low frequency range followed by a plateau at much higher
frequencies (> 1 0 M Hz).
The Coupling M odel
The coupling model developed by Ngai [72] is based on general principles
applied to relaxations in com plex system s. It invokes the concept o f correlated states
am ong the m obile species. In N gai's approach, correlation betw een ions w ould take place
after a critical tim e, t;. Therefore, the relaxation rate w ould be a constant at very short
tim e. A fter the critical tim e t^, the relaxation w ould slow dow n and the tim e dependent
relaxation w ould set in. which can be expressed by
W (t) = — (— ) p
T, t c
(2.15)
The relaxation is then the K.WW function, w ith t 0 and tc being therm ally activated. The
m odel predicts a pow er law behavior for t>tc (or co<uc).
D iffusion C ontrolled Relaxation M odel
Elliott proposed the diffusion-controlled m odel (DCR) based on an interstitial
m echanism in w hich interionic interactions take place in a pairw ise m anner. This model
predicts a dispersive behavior o f conductivity w ith the slope s~0.5 as a consequence o f
the t 12 tim e dependence o f the diffusive behavior [73].
22
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
The Lattice-Gas M odel
This model developed by M aass and B unde considers the inter-ion coulom bic
interactions [74] using M onte Carlo sim ulations for the diffusion o f charged particles in
structurally disordered structures. As a consequence o f coulom b interactions, pronounced
backw ard correlations occur am ong subsequent hops o f ions, which lead to a dispersive
behavior o f the conductivity with a sub-linear frequency exponent. T heir sim ulations
suggest that the com bined effect o f disorder and coulom b interaction provides a
m echanism for understanding non-Debye relaxation.
2.3.4 ADYVP M odel for the NCL Low T em perature C onductivity
The third term in Eqn. 2.4 describes the a in a low tem perature regim e (typically
4K.<T<200K), and low frequencies same as for the U D R region. N ow ick et. al conducted
the first study o f cr in this region on sodium trisilicate glass [52] and proclaim ed it as a
second universality. This region called the 'nearly constant loss' region is characterized
by approxim ately linear frequency dependence w ith w eak therm al activation as show n in
fig. 2.9. Its origin is considered to be totally different from the conductivity in the dc or
L D R region. The theoretical aspect o f the low tem perature s=1.0 conductivity behavior
has been quite extensively studied using an asym m etric double-w ell potential (ADW P)
m odel originally proposed for nuclear spin relaxation at low tem peratures [51,76,77], The
central hypothesis o f the ADW P m odel is based on the assum ption that a certain num ber
o f atom s or group o f atom s occupy one o f the two equilibrium positions o f an asym m etric
double well potential. The group o f atom s is usually m ade o f the netw ork formers and
23
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
m odifier atoms; the latter species, w hen present, are likely to dom inate the overall charge
fluctuations. The double w ell is characterized by an asym m etry energy. A, and average
barrier height energy, V, both o f w hich m ay have a distribution due to the disorder in the
structure and com plex interactions am ong atoms. Under therm al activation, this group o f
atoms can go over the barrier from one potential well to another.
According to the m odel, the total .ADWP conductivity can be obtained by
sum m ing up contributions from all .ADWP configurations in an unit volum e, given by
fS]:
'
a(ft).T) =
^V
R
f fsec h : (A / 2kT)«b(A, V ) — — — dAdV
12kT
JJ
(2.16)
1 +- ( c o t ) ’
w here N is the .ADWP concentration per unit volum e dependent on glass com position. R
is the separation betw een two wells. Vm and Am are the m axim um energy values o f
barrier height and asym m etry respectively, <j>(A,V) is the distribution function, and
r= -,)sech(A 2kT)exp(V /kT) is the relaxation time.
An im portant prediction o f the .ADWP m odel (Eqn. (2.16)) is that a should peak
with increasing T at any fixed frequency. Indeed it has been experim entally observed at
- 100K and verified in several low conductivity germ anate glasses and som e fluoride
glasses glasses, in w hich the contribution from ion hopping at low T is not dom inant
enough to overw helm the peak [51,76]. The rapid increase o f cr above 100K. is
established to be due to the hopping o f ions. Perhaps the success o f the m odel lies in the
fact that it can explain the observed decrease in ionic conductivity w ith increasing
24
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
tem perature beyond the peak in the low alkali germ anate glasses. How ever, physical
description and characteristics o f A D W P structures is still far from clear.
2.3.5 Ionic C onductivity at M icrow ave Frequency
Further increase o f <a to -G H z frequencies causes the conductivity to increase
m ore rapidly than seen in the U D R region, w ith a slope alm ost close to unity (see fig.
2.S). This observation o f the high
frequency
linear behavior is rem iniscent o f
conductivity behavior in the NCL region. Sim ilar observation by Durand et al. [79] on
A gl-doped chalcogenide glass and Belin et. al [80]on A g-doped Ag?S glasses have
confirm ed the existence o f the linear region at high frequencies. Due to the sim ilarities in
the conductivity behavior in the two regions, nam ely the low f - low T and high f - high
T. it is com pelling to believe that the m echanism o f conduction in these regions could
have a com m on underlying origin.
E a r ly T h e o r ie s
Early studies o f the m icrow ave conductivity, aviw, o f oxide glasses were
conducted at one or two frequencies and at room tem perature beginning with the work o f
N avias and G reen, [81] von Hippel [82] and Stevels [83] som e fifty years ago. From such
lim ited m easurem ents f i v e independent sources were proposed for the dielectric loss in
glasses and glass-ceram ics at M W frequencies: [84 - 90]
(i) M igration loss, w hich arises from the transport o f m obile alkali cations, and is
com m only observed as frequency dependent conductivity (in the kH z-M H z region) after
25
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
the dc conductivity plateau in log cr vs log frequency plots. Isard [91] suggested that the
low frequency relaxation arising from alkali ion m igration m ust contribute to losses at
MW frequencies irrespective o f the details o f the hopping process. It is found that for a
sodium tetrasilicate glass above 100°C the observed M W frequency loss is in reasonable
agreem ent w ith the m igration loss as extrapolated from low frequency conductivity data
[S4], However, for a series o f sodium alum inosilicate glasses the extrapolated m igration
loss consistently underestim ates the MW loss, so an additional loss m echanism m ust be
postulated to explain the difference [88].
(ii) Vibration loss from the localized vibrations o f the m obile cations or som e m assive
units, which are characterized by a resonance absorption peak [83]. Localized vibrations
o f isolated alkali cations (in alkali silicate glass, for exam ple) about their sites in the glass
structure are com m only identified by a resonant absorption peak at far infrared
frequencies [92], The extrapolated magnitude o f conductivity (w hich varies x co") as well
as the T dependence from this source is m uch sm aller than the observed values at MW
frequencies [84.88], Therefore, to explain these discrepancies, Topping and Isard [88]
proposed that the T dependence arises from m igration loss. Furtherm ore, the large
m agnitude o f conductivity was suggested as due to resonant vibrations o f large atom ic
groupings at MW frequencies rather than that o f isolated ions.
(iii) D eform ation loss due to sm all deform ation o f the glassy netw ork. It is sim ilar to
dipolar loss but w ith low activation energy (<0.1 eV), and hence presum ed to involve
localized deform ation o f the glass network. Stevels, [93] having proposed loss peak from
this m echanism at low tem peratures (<100 K.) and kH z frequencies, suggested its
26
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
contribution to also dom inate at M W frequencies and RT [85]. D ue to their large
deform ability (polarizabilitv) non-bridging oxygens (N B O ) w ere assum ed to be the
deform ing
species
[86].
However, the
m easurem ents
on
aium inosilicate
glasses
contradict this m odel: the observed loss increases as NBOs decrease w ith the replacem ent
o f Si by Al [8 8 ],
(iv) Electron hopping loss which is sim ilar to the m igration loss but is o f an electronic
origin. It was proposed to explain appreciable loss in alkali-free glasses, but very quickly
discarded as a source o f significant loss in com m on ionic glasses [87-89, 94],
(v) Interfacial polarization loss which, in principle, is possible in glass-ceram ics
containing residual glassy phase. However, considering that the interfacial polarization
has a relaxation frequency in the Hz - kHz region, it cannot be significant at the MW
frequencies. Also, the observed perm ittivity is too sm all to result from any interfacial
phenom enon.
In short, we conclude from the past work that qualitatively one could consider the
m icrow ave conductivity,
c m w
,
o f glasses in term s o f the m igration, the vibration and/or
the deform ation loss m echanism s. In general, the first two m echanism s do not explain the
observed losses quantitatively, and there is no satisfactory m icroscopic description o f the
third m echanism . Thus, in spite o f the plausible three sources, the m icroscopic origin o f
ctmw has rem ained unknow n for glasses.
27
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
R e c e n t M o d e ls
There have been few focused studies to further clarify the origin(s) o f
ctmw
in
glasses after the early work described above. Interestingly, the recent progress has
resulted m ostly from a better understanding o f the low frequency U D R conductivity.
C onsequently, two different m odels o f ctmw have em erged:
(i) The first m odel, principally advocated by Funke et al.[95,96], view s
ctmw
as sim ply an
extension o f the low frequency dispersion o f conductivity at room tem perature and
above. Any additional contribution is ascribed to the tail o f vibration loss peak o f the
cations in the far infrared region, o r som e dipolar loss. This view is sim ilar to the early
explanations described above, but w ith a better description o f the m igration loss o r UDR.
Now the ion hops, but with a tim e dependent coupling w ith its environm ent [71.97],
M icrowave conductivity is then sim ply a short tim e observation o f the relaxation
associated w ith the hopping o f ion. .An im portant feature o f the m odel is the expectation
o f a plateau in conductivity spectrum starting at m icrow ave frequencies and beyond.
Such a plateau has been observed in a few fast ion conducting crystals but its presence in
glasses is not clear [95].
(ii) The other current model treats
ctmw
to have a different origin than the UDR
conductivity [98,99], In this case a Mw arises from localized m ovem ents o f a group o f
atom s sim ilar to that o f a ‘je lly fish ’ in the ‘glassy o cean’. Hsieh and Jain [100] have
dem onstrated a com m on origin for the room temperature-OMw, and low frequency
(-k H z)-lo w tem perature (<200K) conductivity o f a lithium silicate glass. If this
suggestion is generally confirm ed, we can apply all the know ledge about low frequency28
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
low tem perature conductivity for describing crMw- M athem atically the m otion o f jellyfish,
as observed in ac electrical conductivity or nuclear spin relaxation, can be described
reasonably well in term s o f therm ally activated change in its configuration over an
asym m etric double well potential (ADW P) energy barrier [51,101], System atic studies on
polycrystalline ceram ics such as G d3^ and Y'^ -doped CeCN by N ow ick et al. [ 102] have
strongly supported the ADW P concept. They report, w ith increasing charge carrier
concentration, a gradual change o f Debye type dipolar dielectric loss peak into ADW P
type power law [103],
For a long tim e the structure o f a jellyfish configuration rem ained a mystery, but
recent system atic studies on series o f glass com positions have revealed significant details
about
the
constitution
o f these
entities
[51,101,104],
For exam ple,
in
lithium
silicophosphate glass, three kinds o f jellyfish structures have been identified from the low
tem perature data. Therefore, differences are possible betw een the jellyfish structures
dom inating at different tem perature and frequency regim es.
Having thus understood som e o f the early conduction m odels and new theories, it
becom es im perative to understand which m echanism is responsible for m icrowave
conduction in glasses.
29
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
G i >2s a
A
i
i
Tt m p e r a t u r e
---------------- ►
Figure 2.1: Specific volum e versus tem perature curves for glass formation [105],
30
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
o
0
o
0
f *
0
Ch T
^
O
z
o
o
°~ G
d
C
^
4
^
^
S_J
x y *
> < ^ o
P
o
~ "o
o
^ -Q
- vJ
C
0
0 ^ 0
£ ° %
> < H
K H
H
%
c2 o ^
c r\> 4
<P
p
o
o
fa)
(b)
C ry stallin e , (b) Glassy AtO, c o m p o u n d [1051.
Fi<mre 2.2: Structure o f (a)
31
■ -on o ttn e copytigm owner. FurtPer reproduction p r o P « e d wi<Pou, perm ission.
R eproduced with perm ission of the copy y
(a)
— Si— O — S i — + M tO = — Si— ( T N T
M ^ ' O — Si-
(b)
• Si
o
0
Na
Figure 2.3: (a) M odification o f silicate netw ork upon addition o f N a ions, (b) structure o f
sodium silicate glass [105].
32
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
• F orm ation of 6-fold coordination of G e atom
i0
1
10
I
0A
I0
- 0 —G e —O —G e —O — + MJD
I
=
01
I
0I
I
/ M'\
- O —Ge —O
0 — Ge — 0 —
0'
I
M •
I
“
01
I
(4-fold ♦ 2NBO)
0
I
0
I
O —Ge — O —G e —O — * M.0
I
0
=
I
0
I
0
I
o
I-/
—0 — Ge — 0 — G e — 0 -
I
0X
0 /
■
1
y
m
■
(6-fold, no NBO)
Figure 2.4: Form ation o f 4-foid and 6-fold coordinated G e atom s in binary germ anate
glasses due to the addition o f alkali oxide [30].
JJ
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
0.05
25%LioO 75%BoO
0.04
178.5 C
E
o
CA
1
132.1 C
0.03
>
,62.5 C
>
o
=3
0.02
■a
ro
O
132.1
0.01
0.00
co (rad/sec)
Figure 2.5: C onductivity, or and electric m odulus, M " as a function o f frequency for
lithium triborate glass.
34
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Ion bop
o
( ^
^
) —
0
E
II
I
[
r
r
Figure 2.6: A nderson-Stuart M odel o f ionic m igration in alkali silicate glasses [56].
35
R e p r o d u c e d with p e r m i s s io n of t h e cop y rig h t o w n e r. F u r th e r r e p ro d u c tio n prohibited w ith o u t p e rm is s io n .
Figure 2.7: Haven Ratio, H r. as a function o f N a?0 concentration in sodium borate glass
[63].
36
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
10J
10:
Na20 .3 S i0 2
10'
10°
10"’
E
.o
CO
>
>
o
13
■a
c
o
O
10'2
10'3
10 "
10'5
;OCOC«*XCC‘XOC'5<*<X
10 "
9
10 ‘ :
10 "
10'9
10
■
T=0 C
0
T=25 C
V
T = 1 10 C
X
T=168C
A
T=268 C
T=483C
.1 0
10 '
J
10 '
-2
2
4
6
i
8
L
10
J
12
14
i_
16
Iog(frequency, Hz)
Figure 2.8: Conductivity versus frequency plot o f sodium trisilicate glass at different
tem peratures [65],
37
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1 E —6
1 E —7
1E -8
142
-
E
°
c:
1 E —9
IE -
tE-
1E—
IE 10
100
1000
1E 5
F requ en cy (Hz)
Figure 2.9: C onductivity versus frequency plot o f sodium trisilicate glass show ing NCL
region betw een 4K and 142K [52].
38
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CHAPTER 3 OBJECTIVES AND OVERALL APPROACH
3.1 O B JE C T IV E S
From the assessm ent o f the current literature, we can state the m ost significant
issues pertaining to ion dynam ics in oxide glasses at m icrow ave frequencies in the form
o f the follow ing statem ents and questions:
1) The prim ary objective is to establish the m echanism o f m icrow ave conduction in
oxide glasses.
2) To verify if the low frequency - low tem perature conductivity and high frequency high tem perature conductivity have the sam e origin. If so, can they both be described
by the sam e A D W ? configurations?
3) W hat is the physical description/characteristic o f the A D W PC s that are responsible
for m icrow ave conduction?
3.2 O V E R A L L APPRO AC H
O ur approach to answ er objective (1) is to obtain a d ata for selected glass system s
o f varying com position at m icrowave frequencies as a function o f tem perature. The
choice o f the glass to serve as a m odel system partly depends on the conductivity levels at
such high frequencies. Typically the glasses should possess m edium to high conductivity
values (>1CT S/cm ) in order to perform m icrow ave m easurem ents w ithin the capabilities
o f the experim ental set up.
39
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
T o verify if the A D W P m odel is applicable for the low f - low T and high f - high
T regions, the respective data w ill be analyzed according to the A D W P form alism (Eqn.
2.16).
If com parable sets o f fitting param eters, nam ely, N, the A D W P concentration;
V m, the m axim um (cut-off) energy for barrier height; Am, the m axim um for the
asym m etry energy, are obtained, then our hypothesis that the conduction in the two
regions arise from the sam e A D W P configurations w ill be proven valid. By contrast,
large differences in the param eters will be an indication o f different kinds o f AD W PCs
for the two regions. T he latter possibility is not unlikely, since different AD W P structures
are possible at different tem perature and frequency regim es. At this point, we assum e the
sim ple expression for the distribution function O(A.V) (in Eqn. 2.16) used in the
literature.
It is how ever im portant to establish the m athem atical form o f <j)(A,V) if the
A D W P is to serve as a fundam ental model o f ion conduction. O n the other hand, if high f
high T region cannot be described by the ADW P form alism , then we m ust seek an
alternative m echanism or a new theory o f m icrow ave conduction in glasses.
The choice o f the glass system is particularly crucial for answ ering objective (3).
Based on the inform ation in literature and our recent studies at low T and low f on several
oxide glasses, it is know n that variation in the com position o f glass causes structural
changes and large variations o f the ADW P param eters. A lso, keeping in m ind the
requirem ent o f high a for VIW m easurem ents, we propose to investigate at least two
glass series as m odel system s. O ur first choice o f glass system w ould be the lithium
silicate glass, for w hich o data at low f and low T have already been extensively studied.
The second glass series w ould be the potassium germ anate series, for w hich detailed
40
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
structural inform ation from different spectroscopic techniques (XPS, IR, EXAFS, NM R
etc.) is available. The low f and low T conductivity for this glass series has also been
studied previously in our group. W ithin each series, a specific aspect o f glass structure
will be varied system atically. By correlating the ion dynam ics to the m aterial structural
param eters, a physical description o f the A D W PC s will be obtained.
41
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CHAPTER 4 EXPERIMENTAL TECHNIQUES
The goal o f this w ork is to seek answers to the questions stated in chapter 3 by
invoking the following experim ental tools and techniques.
4.1 G L A SS P R E P A R A T IO N
The glass com positions prepared for the present investigation arc lithium silicate
glass o f com position S b L iiO o A h O j-lP ^ O s-d lS iO : and potassium germ anate glasses,
x K :0 -(l-x )G e O :, 0.23<x<25 mol% . The glasses are prepared b y the conventional meltquench technique.
4.1.1 L ith iu m Silicate G lass
Lithium silicate glass o f the above com position w as prepared at Sandia National
Laboratories by thoroughly m ixing appropriate am ounts o f puratronic grade o f 99.999%
purity LLCO.', AI2O 3, S i0 2 and P2O 5 powders in a roll m ill for about 24 hours. The
thoroughly mixed pow ders are then heated to about 1400°C in a high tem perature
induction furnace. The m elt is allow ed to equilibrate for about an hour at the m elting
tem perature and then quenched to room tem perature (RT) b y pouring the liquid into
stainless steel mold. T he glass form ed is then annealed at - 6 0 0 ° C for about 1 hour to
rem ove any internal stress that m ay have form ed during the rapid quenching. The
annealed glass is then furnace cooled to RT and then cut into thin plates o f 1cm x lem x
1mm and deposited w ith Au electrodes for low frequency m easurem ents. For MW
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
m easurem ents, the glass was precision m achined b y B orges T ech+, PA to fit a coaxial
airline.
4.1.2 .tK i0 -(l-.x )G e 0 2 Glass Series
A ppropriate am ounts o f puratronic grade o f 99.9999% purity G eO : and 99.999%
purity K :C 03 pow ders are thoroughly m ixed for about 24 hours in a roll m ill. The
pow der m ixture is preheated to about 1000°C for 1-2 hours for the decom position o f CO :
gas and then m elted in a platinum crucible. The m elt is equilibrated for about an hour at a
tem perature betw een 1200 and 1600°C depending on the com position and viscosity. The
hom ogenized melt is quenched to RT by pouring the m olten liquid into stainless steel
m old. T he glass formed is then annealed for Vi hour at -4 5 0 -5 0 0 °C in an annealing
tum ace to rem ove internal stress. A fter annealing the glass is furnace cooled to RT and
then shaped to the required geom etry for electrical m easurem ents.
4.2 A U D IO & RF EL EC TR IC A L M E A SU R E M E N T S
4.2.1 Principle
T he audio and radio frequency electrical m easurem ents typically involve the
m easurem ent o f the capacitance, C and the conductance, G o f the glass sam ple in the
frequency range o f 10 Hz to 100 kH z and tem perature from 4 K to room tem perature. The
sam ple is generally treated as a lum ped-circuit w ith finite resistance and capacitance
' Borges Technical Ceram ics, Pennsburg, PA
43
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
values. The m easurem ents are done using a capacitance bridge, w hich works on the
principle o f a W heatstone-type bridge w ith the sam ple connected to one arm o f the
netw ork and with transform er ratio arm s as show n in fig. 4.1 [106], T he ac signal from
the generator (GEN) w hen applied to prim ary w indings, Np, induces voltages, Vs and Vx
in the secondary w indings o f ratio Ns/Nx, w here N s and Nx are the num ber o f turns in the
two w indings respectively. W hen the bridge is balanced and no current flows through the
detector (DET),
G
N
-icoC ^ = — - ( G
Nt
H-icoC.)
(4.1)
where u (=2rcf) is the angular frequency o f the ac signal. Thus the values o f conductance
(C-J and capacitance (C x) o f the sam ple can be determ ined in term s o f the standards, Gs
and C,.
4.2.2
Experim ental set-up
The capacitance and conductance o f the glass sam ple w ere m easured using a
General Radio 1616 Capacitance Bridge, w hich is capable o f m easuring in the frequency
range o f 10 Hz to 100 kHz. The glass sam ple in the form o f thin plates/discs is deposited
with a thin film o f Au metal, which acts as electrodes. In order to m inim ize conduction
through the surface o f the sample, it is necessary to keep the surface conduction path
greater than the volum e conduction (thickness) path [107], In the sim plest case, called
the tw o-probe m ethod, electrodes m ay be deposited on both surfaces o f the thin sample.
How ever, to avoid the effect o f short-circuiting o f the tw o surfaces, three-probe m ethod
is often preferred as show n in fig. 4.2a. H ere a guard ring is placed betw een the central
44
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
electrode on the top surface; any current betw een the guard ring and the bottom electrode
is excluded from the m easurem ent. A nother advantage o f using the guard ring is the
elim ination o f uncontrolled fringe effects that distort the field distribution, m aking it nonuniform . Here, the effective electrode area is given by,
A d ec. = T t W
( 4 -2 )
w here rclr is:
r.,r = r , + f - — I n c o s h ( ^ )
2
;c
4d
(4.3)
with n as the central electrode radius, g the gap w idth and d the sam ple thickness [108].
The last term in the above equation is a small correction term and is often neglected.
M easurem ent o f C and G o f the sam ple are obtained as a function o f frequency
and tem perature. The conductivity,
cr, is then calculated by m ultiplying
G with the
geom etric factor, d/A, o f the sample. The dielectric constant (s') and loss (s") can also
be
calculated using the following equations:
s' = —
e .A
(4.4)
s" = —
(4.5)
COE ,
w here e0 is the perm ittivity o f vacuum.
4.3
M IC R O W A V E ELEC TR IC A L M E A SU R E M E N T S
M icrow aves are electrom agnetic radiation w hose frequency typically cover the
range from 109 Hz (w avelength,
a.
= 30 cm) to 10u H z (X = 0.3 cm). Since the
45
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
w avelengths are small com pared to som e physical dim ensions o f the sam ple under study,
two closely spaced points on the device can have significant phase difference. As a
consequence, the m easurem ent techniques at high frequencies are very different from
those at low frequencies because sim ple com ponents, such as a connecting wire, which
behaves perfectly at low frequencies, behave differently at high frequencies.
4.3.1
E le ctro m a g n e tic W ave P ro p a g a tio n
The electrom agnetic theory is best presented starting from M axw ell’s equations:
VxE =
(4 6 )
ct
VxH=J~ —
ct
(4.7)
V.D - p
(4.8)
V.B - 0
(4.9)
where E is the electric field intensity, D is the electric flux, B is the m agnetic flux, H is
the m agnetic field intensity, J is the current density and p is the charge density. The
electrom agnetic field vectors, E, D, B and H are related to each other by the following
relations.
D = sE
(4.10)
H =—
(4.11)
where s and q are the perm ittivity and perm eability o f the m edium respectively. A t the
interfaces, the field vectors are continuous. M axw ell’s equations above hold for
46
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
electrom agnetic quantities w ith arbitrary tim e dependence. T im e harm onic (steady-state
sinusoidal) fields occupy an unique position since they are easy to generate and they can
form
fields w ith arbitrary periodic tim e functions according to the principle o f
superposition. For exam ple, one can w rite a tim e harm onic field referring to a cosine
function as
E (x. y, z, t) = Re[E(x, y, z )e J“l j
(4.12)
w here to is the angular frequency o f the field, and x, y and z are the space coordinates. In
a sim ple (linear, isotropic and hom ogeneous) m edium , tim e harm onic M axw ell’s
equations can be w ritten in terms o f the field (E and H) and source (J and p) phasors as:
V xE = -jo )(iH
(4.13)
V xH = J
(4.14)
t
jcosE
V.E = —
e
(4.15)
V.H = 0
(4.16)
Eqn. (4.14) can be reduced to
VxH = c tE + jcoeE = jcoE(e - j —) = jco e'E
G)
(4.17)
if one defines the com plex perm ittivity as s*=e'-je", w here s '= e and e"=<t/ g). Furtherm ore,
if the free charge density in the m edium is zero, then Eqn. (4.15) becom es
V.E = 0
(4.18)
C om bining Eqns. (4.13, 4.16, 4.17) and (4.18) for a charge-free sim ple m edium and
taking curl o f Eqn. (4.13) to yield a second-order partial differential equation in E,
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
(4.19)
V -E + co-Lie E = 0
In a sim ilar w ay, one gets
(4.20)
V 2H + e r u £ ’ H = 0
Eqns. (4.19) and (4.20) are know n as hom ogeneous vector H elm holtz’s equations. One
can define the propagation constant, y as
(4.21)
w here a and p are the attenuation and phase constant respectively. Therefore, Eqns.
(4.19) and (4.20) can now be w ritten as follows:
V : E - y ;E = 0
(4.22)
V - H - y H =0
(4.23)
4.3.2
W ave p ro p a g a tio n in a tra n sm issio n line
As show n in fig. 4.3a, a transm ission line is often schem atically represented as a
tw o-w ire line, since transm ission lines for transverse electrom agnetic (TEM ) propagation
alw ays have at least two conductors. A short piece o f a line o f length Az (in fig. 4.3a)
can
be m odeled as a lum ped-elem ent circuit as shown in fig. 4.3b, w here R, L, G and C are
p er unit length quantities defined as follows.
R = series resistance per unit length (for both conductors) in Q /m .
L = series inductance per unit length in H/m.
G = shunt conductance per unit length, in S/m.
C = shunt capacitance per unit length, in F/m.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
The series inductance, L, represents the total self-inductance o f the two
conductors and the shunt capacitance, C, is due to the close proxim ity o f the two
conductors. The series resistance, R, represents the resistance due to a finite conductivity
o f the two conductors, and the shunt conductance, G, is due to the dielectric loss in the
m aterial betw een the conductors. R and G, therefore, represent loss. A finite length o f
transm ission line can be view ed as a cascade o f sections in the form o f fig. 4.3b [109],
From the circuit, the voltage and the current can be determ ined by applying
K irc h o ff s laws to give [109],
v(z. t ) - RAzi(z, t) - LAz °^- Z: ■ - - v(z + A z,t) = 0
a
i(z, t)- G A z v ( z + Az, t ) - C A z
cv(z + Az. t)
a
- i ( z + Az, t) = 0
(4.24a)
(4.24b)
D ividing Eqns. (4.24a and b) by Az and taking the lim it as Az—>0 gives the following
differential equations:
(4.25a)
a
cz
ci(z, t)
(4.25b)
cz
These equations are the tim e dom ain form o f the transm ission line. For the sinusoidal
steady-state condition, with cosine-based phasors, Eqn. (4.25) sim plify to the following
equations.
dV (z)
= - ( R + jcoL)I(z)
(4.26a)
dz
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
= - ( G + jcoC)V(z)
(4.26b)
dz
Here v(z) and i(z) are the real com ponents o f V fzje1” ' and I(z)e*ft)t respectively. The above
two equations can be solved sim ultaneously to give w ave equations for V(z) and I(z):
d : V (z)
■ -y 'V (z ) = 0
(4.27a)
dz:
~ ~ r ~ ~ 7 : i(z) = 0
dz'
(4.27b)
where y is the propagation constant given as:
7 = a + j p = tJ ( R + jw L )(G + jcoC)
(4.28)
T ravelling wave solutions to Eqns. (4.27) can be w ritten as:
V (z) = 'V0 ' c " rz + V 0 ~ e T‘
(4.29a)
l(z) = [ / e ^ + [ 0 e ^
(4.29b)
w here the term e 'rl represents the wave propagation in the +z direction and
represents
the propagation in the - z direction. W e define the characteristic im pedance, Z0, such that
=
IRHJ5 L
7
(430)
^ G + jcoC
In a sim ple case w here the transm ission line is ‘loss-free’, then setting R=G =0, we get
7 = a + jp = jcoVLC
(4.31)
im plying a = 0 and P=co(LC)l/1. The characteristic im pedance reduces to Z0= (L /C )l/:,
which is now a real num ber. Generally, in a lossless line, Zo=50Q .
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R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
4.3.3 P rinciple o f S-param eter m easurem ent
W ith the advent o f the m odem instrum ents to m easure dielectric perm ittivity o f
m aterials
at
high
frequencies, broadband
m easurem ents
have
gained
popularity.
Typically, the behavior o f the device under the influence o f electrom agnetic radiation
(here, m icrow aves) is studied by instrum ents called vector netw ork analyzer (VNA).
N etw ork analyzers are instrum ents that m easure im pedances and/or transfer functions o f
linear netw orks through sine w ave testing. A VN A system
accom plishes these
m easurem ents by configuring its various com ponents around the device under test
(DUT). The netw ork analyzer consists o f a sine w ave source to stim ulate the DUT. The
device/m aterial interacts with the radiation absorbing a part o f it, w hile the rest is either
reflected and/or transm itted. The reflected and transm itted signals are separated and
detected by the detector. A schem atic representation o f a V N A is show n in fig. 4.4.
O ne o f the com m on netw ork configurations is the tw o-port netw ork as shown in
fig. 4.5a. T he DUT is connected betw een the tw o ports o f the netw ork analyzer. The
reflected and the transm itted signals are m easured in the form o f scattering param eters,
com m only referred to as S-param eters. S-param eters are ratios o f the reflected and
transm itted travelling w aves m easured at the two ports o f the analyzer. T he reflected
signals at port I and port 2 are designated as S n and S 22', the transm itted signals are
designated as S21 and S \ i (1 and 2 represents the respective ports) as represented in the
flow graph fig. 4.5b. T he notation ‘a f in the figure refers to the signal at the input end o f
the device w hile ‘b j ’ refers to the signal at the output end. For a tw o-port device, the
expressions o f the S-param eters are given by: [110]
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
: r g -z :)
(4.32)
i-(rz )-
(4.33)
(4.34)
where R| and Ri are the respective reference plane transform ations given by R |,: = exp(YoL i .;); y„ being the propagation constant in an em pty airline, Li and Li being the
distances from the calibration reference planes to the sam ple ends. In the expressions
(4.32 - 4.34), z represents the transm ission coefficient and f represents the reflected
coefficient, both o f which are related to m aterial param eters in the following manner.
z = exp(—yL)
(4.35)
f =
(4.36)
Here L is the length o f the sample, p* and s ' are the com plex m agnetic perm eability and
perm ittivity respectively, Cvac and C|ab are the velocity o f light under vacuum and
laboratory conditions. Since R u , L i^ are unknow ns and cannot be m easured accurately, it
is necessary to com bine all the four S-param eters to yield an equation w ith m inim um
unknowns, w hich is expressed as:
(
l
j~2 \
(4-37)
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
where Lair, the length o f the em pty airline, and L, the length o f the sam ple can be
accurately m easured. Thus the left-hand side o f Eqn. (4.37) is m easured by the VNA,
w hich is related to the m aterial param eters, p* and s* on the right-hand side.
4.3.4 E x p e rim e n ta l set-up
The experim ental arrangem ent show n in fig. 4.6a consists o f a netw ork analyzer
HP8753C which is connected to a S-param eter test-set H P85047A that is capable o f
m easuring the S-param eters in the frequency range o f 50M H z to 3GHz. O ur experim ental
set-up
allow s m easurem ents
from room
tem perature to about
120°C
using the
transm ission/reflection method. The transm ission line consists o f a stainless steel coaxial
airline o f length 60.96 mm (2.4 in), the ends o f which are term inated by standard APC
connectors o f impedance 50Q . The coaxial line was precision m achined by A stro la b \ NJ.
The coaxial airline (fig. 4.6b) conform s to standard specifications, consisting o f two
concentric conductors, the outer conductor o f diam eter 7 mm (0.275 in) and the center
conductor o f diam eter 3 mm (0.118 in). A detailed description o f the inner (or center) and
outer conductor that constitutes the coaxial airline are given in A ppendix A. Glass
sam ples o f lithium silicate o f sam e com position with varying lengths 7.37 mm and 19.72
mm, and potassium germ anate glasses o f different com positions w ith lengths ranging
betw een 10.74 mm to 12.59 m m were precision m achined to fit the coaxial airline w ith
negligible air gap betw een the glass sam ple and the conductors. T he glass sam ple serves
as the device under test (DUT). The reported m icrowave conductivity results are found to
f Astrolab Inc., New ton, NJ.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
be independent o f the sam ple length, and thus confirm the reproducibility o f the
m easurem ents w ithin ±7% .
T w o-port calibration is done prior to the actual S-param eter m easurem ent, using
the 7 m m calibration kit HP85031B, to m inim ize instrum ent errors arising from port
cables, adapters, connectors etc. The sam ple is then placed inside the coaxial airline and
the S-param eters are m easured as a function o f frequency. A cquisition o f data is done
through an autom ated system . Analysis o f the m easured data using Eqns. (4.32 - 4.37) is
done through an iterative program ‘E PSM U 3’ developed by Baker-Jarvis [111], which
utilizes N ew ton-R aphson algorithm to calculate the dielectric perm ittivity from the Sparam eters. K now ing the dielectric loss, the conductivity is calculated using the relation
CT-co£oe"
(4.38)
The m easurem ent is then repeated at different tem peratures. A step-by-step instruction to
m icrow ave m easurem ent, data acquisition and dielectric perm ittivity calculations using
E PS M U 3’ is described in Appendix B for further clarity.
4.4 FAR IN FR A R ED (FIR ) M EA SU R EM EN TS
R adiation absorption at very high frequencies in the IR and Far IR regions can be
exploited to obtain inform ation about the structure o f a substance.
These regions o f
absorption depend on the interatom ic forces and structural arrangem ent o f the constituent
atom s, w hich in turn, affect the vibrational m odes. V ibrational spectroscopy has been
em ployed ov er the last several decades to investigate the structure o f glasses. Infrared
spectroscopy, in particular, is proven useful because it provides a m eans to determ ine the
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
local structure w hich constitutes the glass netw ork, som e properties o f the sites hosting
the m odifying cations, and interactions betw een the charge carriers and the netw ork units
[ 112 ].
Because glasses are strong absorbers o f infrared (IR) radiation, m ost o f the early
spectra w ere obtained by transm ission o f pow dered glass dispersed in a suitable m atrix,
e.g.: alkali m etal halides. H ow ever this technique often led to spectral distortions and to
non-reproducible intensities o f absorption bands, m ainly because such spectra originate
from a com bination o f transm ission and reflection phenom ena [113]. As a result, the
m easured phonon frequency does not coincide w ith that o f the infrared active transverse
optical (TO) m ode as it should. M oreover, the addition o f alkali halide salts as m atrix
m aterials lead to partial hydrolysis o f the glass sam ple [114] and in som e cases even, ion
exchange at high pressures [115].
One o f the advantage o f infrared reflectance spectroscopy is the use o f the same
sam ple over a broad frequency range covering both the m id- and far-infrared. This
advantage
is
com bined
with
the
capabilities
o f the
m odem
Fourier-transform
spectrom eters, and the availability o f softw are for the proper analysis o f the reflectivity
data. Therefore, the true bandshapes o f the transverse optical (TO ) and longitudinal
optical (LO) m odes are obtained, and this allows for the quantitative analysis o f the glass
structure. These features m ake infrared reflectance spectroscopy a pow erful tool in
probing glass structure and dynam ics.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
4.4.1
A nalysis o f [R reflectance data
The experim ental reflectivity data for a sam ple o f sem i-infinite thickness can be
analyzed by two different procedures: (a) the Kxam ers-K ronig inversion without
reference to any model o f dielectric response, and (b) reflectance curve fining using a
theoretical m odel for the dielectric function. In the K ram ers-K ronig (K-K.) method, the
m easured specular reflectivity R(v) is transform ed to calculate the phase angle S(v)
betw een the reflected and incident wave [116]:
S(V) = ^
‘f l n r ( v '. > - |nl W dV|
* ;
( 4.39)
v -v,-
where r(v)= (R (v))!
Extrapolation o f the reflectivity data to v—>0 and v—»oc is done
using an appropriate software. The spectrum o f 3(v ) is then utilized to obtain the real
(n(v)) and the im aginary (k(v)) parts o f the com plex refractive index (n ‘(v)= n(v)-ik(v))
on the basis o f the Fresnel formulae for norm al incidence:
n(v ) = ----- ;—
------------------------------------1 - r ‘ (v) - 2r(v) cos 3(v)
k( v) =
_ 2 rM s in S (v )
(4.40)
(4 4 1 )
1 + r" (v) - 2r(v) cos 9(v)
C om putation o f the real ( e'( v )) and im aginary (s"(v)) parts o f the com plex dielectric
perm ittivity (e*(v)= e'( v )-h e "( v )) is then straightforw ard:
e '( v )
= n : ( v ) - k : (v)
(4.42)
E"(v) = 2 n (v )k (v )
(4.43)
56
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
In analogy to crystalline solids, m axim a in e"(v) give the TO frequencies, while
corresponding m axim a o f the energy loss function, -Im (l/e (v)) give the LO frequencies
[117]. In m any cases, the results o f the K.-K. analysis are expressed in term s o f the
absorption coefficient, a (v ), calculated by:
a ( v ) = 4iivk(v)
(4.44)
If additional inform ation is required on the param eters o f the spectral bands, such as
resonance frequency, bandwidth, intensity etc., the absorption coefficient spectra can be
resolved to com ponent bands by using standard non-linear square fitting techniques
[118],
The alternate m ethod o f analyzing reflectivity data is the reflectance curve fitting
procedure. A ccording to the classical dispersion theory, the com plex dielectric function,
e'(v> of glass is m odeled as follows [119]:
e ' ( v)
AE v
= e ’( v ) -r ie"(v) = £ c f Y — ;-----Lr-!
(4.45)
. Vj'-v'-ivTj
where the sum m ation is over j Lorenztian oscillators, each one characterized by three
param eters: the resonance frequency, v]5 the bandw idth, Tj and the dielectric strength, As,.
Here
is the high frequency dielectric constant o f the glass. The expressions for e '( v )
and s"(v) from Eqn. (4.45) can be written as:
„
S' (V) = E , +
X
j
_
AE:V (v 2 - v : )
—
----------- 7
( v / - v - ) - + (T ]V)-
(4-46)
A s,v T v
g"(v) = X .
]
(Vj
- V
T - .V -TT
)
(4-4?)
^(TjV)
57
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
The param eters vjt Tj and ASj for each oscillator and
are then determ ined by best fitting
the calculated reflectance (Eqn. (4.48)) to the experim ental reflectivity spectrum .
R,v ) . ^ ) - ' ] ; ^ ( v ) ;
(448)
[ n ( v ) + lj - + k ( v ) -
w here n(v) and k(v) are obtained from Eqns. (4.46) and (4.47) using the relations in
(4.42) and (4.43) respectively. N ote that a disadvantage o f the reflectivity fitting m ethod
is that the absorption coefficient, a (v ) cannot be obtained w ithout assum ing the num ber
o f oscillators as input to the fitting procedure.
C om parison o f analysis using the two m ethods (K.-K. analysis and reflectivity
fitting) for K ;0 -2B203 glass system is shown in fig. 4.7 in the frequency range 30 - 3000
c m '1 for the different param eters [120]. The solid lines are results from the K.-K. analysis
while the dotted lines are from the fitting procedure to Eqn. (4.45) using 12 Lorenztian
oscillators. It can be seen that the agreem ent betw een the two spectra is w ithin the width
o f the lines over the full range o f frequency. The sam e w as found to be true for
3 2 .5 N a :0 -6 7 .5 G e 0 : glass [121] and a m ulticom ponent superionic system , .rCuI-(lx){C u20 - M o 0 3} [122], confirm ing that the two m ethods o f analysis give equivalent
results. W hich o f the two m ethods is used depends on the inform ation required in each
case.
4.4.2 E xperim ental set-up
Infrared reflectance data at frequencies > 0.45 T H z w ere obtained at room
tem perature on our glasses by our collaborator Dr. ECamitsos using a Fourier-transform
58
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
vacuum spectrom eter (B ruker 113v) (see fig. 4.8a), equipped w ith an 11° off-norm al
specular reflectance attachm ent (see fig. 4.8b) and a high reflectivity A1 m irror as
reference. Five m ylar beam splitters o f different thickness (3.5-50 urn) w ere used in the
far infrared and a KJ3r beam splitter in the m id infrared to m easure a continuous
reflectance spectrum in the 15-5000 c m '1 range. Softw are provided by Bruker was
utilized first to extrapolate the reflectance data outside the range o f m easurem ents (v—>0,
v —>cc), and then to perform the Kramers-fCronig transform ation. T he obtained optical and
dielectric properties o f the glass were used to calculate the absorption coefficient using
Eqn. (4.44). From the absorption coefficient, the far infrared conductivity was calculated
using
o ( v ) = e„n(v)a(v)
(4.49)
59
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
GEN
OET
Figure 4.1: Schem atic representation o f W heatstone bridge w ith sam ple connected to one
arm o f the netw ork [106].
60
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
(b)
Figure 4.2: Schem atics o f low frequency electrical configuration (a) three terminal
electrode on sam ple, (b) circuit connections [107],
61
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r re p r o d u c tio n prohib ited w ith o u t p e r m is s io n .
Kz. t)
►
+
v(z. 0
Az
(a)
i(z. t)
i(z +Az, t)
►
O
+
v(z, t)
*-
w
w
RAz
/Y Y Y \
LAz
GAz
~r~
CAz
v(z + Az, t)
A z(b)
Figure 4.3: Transm ission line for transverse electrom agnetic propagation (a) short piece
o f transm ission line, (b) lum ped-elem ent circuit [109],
62
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
VNA
DISPLAY
TRANSMITTED
REFLECTED
DLT
Sn
<—
Si:
Figure 4.4: Schem atic representation o f V ector N etw ork A nalyzer. The device-under-test
(DUT) is connected betw een two ports. S n and S 22 are reflected signals w hile S;i and S i;
are transm itted signals.
63
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
(a )
I
INCIDENT
(FORWARD)
•t i n
.
SU
;
R EFLEC TED I
?b 2
—I
fC
ic
■>1
I
PORT
S -17
i
_
I t ----------i
>!
|
i
R EFLECTED
PORT 2
INCIDENT
TRANSM ITTED
s .,
[R E V E R S E )
(b)
'7 2
b,
S-Parameter
S ,,
§21
S12
S22
Teat Set
Description
Definition
0,
□irection
Input reflection
coefficient
FWD
Forward gam
FWD
R everse gain
REV
Output reflection
coefficient
REV
■af - 1 a2 = °
" IT
b2
a2
a' “ °
i a,=*0
Figure 4.5: (a) Tw o-port configuration, (b) flow graph o f scattering param eters.
64
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Network Analyzer HP 8753C
(a)
CD CD CD Q
CD CD CD
HPIB
CD CD CD
© © © @
s r \ .
P art A
S-parameter test set HP 8S047A
Port B
poooooooo
DUT
□
txxxxxxxxJ
Voltage Reacout
If
I Temoerature
^ Controller
(b)
0 .1 2 ' Oia.
0.27* dia.
l: :
2.4-
Figure 4.6: (a) Experim ental arrangem ent o f m icrow ave m easurem ent, (b) geom etry o f
sam ple and coaxial transm ission line.
65
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
0.2
exp
01
-
0.0
K.-K
15
10
0.6
0.0
0
1000
500
1500
2000
W A V E N U M B E R S (cm*1)
Figure 4.7: C om parison between K ram ers-K ronig and reflectivity m ethod for K.2O.2B 1O3
glass system
in the frequency range 3 0 -2 0 0 0 c m '1. T he solid line is the K.-K.
transform ation o f data while the dotted line is the fit from reflectivity m ethod (Eqn. 4.45)
[ 120].
66
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p ro d u c tio n prohibited w ith o u t p e rm is s io n .
(a) sources, (b) aperture, (c) filter, (d) beam splitters, (e) moving mirror, (f) fixed mirror,
(g) reference interferometer, (i.j) sample channels. (1,2) flip mirrors, (k) detectors.
S am ple
R = lr / I,
(b)
Figure 4.8: (a) B ruker 113v Fourier-transform vacuum spectrom eter for far infrared
reflectivity m easurem ents, (b) off-norm al sam ple arrangem ent.
67
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CHAPTER 5 ADWP MODEL AND ITS PREDICTIONS
5.1 O RIG IN AN D C H A R A C TE R ISTIC S O F A D W P M O D EL
In order to interpret the low tem perature properties o f glasses, considerable
theoretical effort has been expended.
T he so-called tunneling m odel or the two-level
system (TLS) m odel developed independently by Phillips [123.124] and by Anderson et
al. [125] was a great success among other m odels for describing conduction by tunneling
at very' low tem perature (<3K). A ccording to the m odel, the peak in the dielectric and
acoustic loss could be analyzed in terms o f a distribution o f relaxation times, with each
reiaxing elem ent, or dipole, moving in a potential.
The TLS m odel assumes that the glass is a m etastable configuration o f atoms, and
a certain num ber o f atoms (or group o f atom s) can occupy one o f (at least) two
equilibrium positions o f a double well potential. The atoms m ove in an anharm onic
potential, such as the double-well potential show n in fig. 5.1, which in general is
asym m etric, characterized by asym m etry energy, A, and an average barrier height, V.
Due to the random ness o f the glassy netw ork, both A and V are said to have a distribution
o f values in the am orphous solid w ith upper limits Am and Vm, w hich are cut-off energies
for A and V respectively; this distribution in V and A results in the distribution o f
relaxation tim e, t. The positional coordinate q in fig. 5.1 is term ed ‘configurational’
coordinate rather than a ‘space’ coordinate in o rder to disguise the unknow n m icroscopic
nature o f the potential profile. At very low tem perature, say T < 3K . the atom or the group
o f atom s tunnels back and forth betw een the two m inima.
A t higher tem peratures
68
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
(3K<T<150K), the atom s com prising the TLS w ill change configuration by m eans o f a
therm ally activated process, resulting in excitations over the energy barrier V.
In this
tem perature range, the TLS is often called the asym m etric double w ell potential (ADW P)
configuration in order to distinguish the tunneling process (at T<3K ) from the therm ally
activated process. Due to their low concentration, the A D W P transitions are assum ed to
be uncorrelated. D espite its great success in explaining the low tem perature ion
dynam ics, the description o f .ADWP is know n only for very few system s [104],
Therefore, the description o f a ‘je lly fish ’ structure could be illustrated with the
following points:
1) It is a group
o f atoms . w hich collectively m oves betw een different
configurations, m uch like the wiggling o f a jellyfish in a glassy ocean.
2) There is no single atom hopping involved.
3) The fluctuations o f the jellyfish structures are m uch sm aller than typical atom
vibrations.
4) The exact nature o f the jellyfish depends on the m aterial.
5) In the sam e m aterial, more than one jellyfish m ight exist and be observed in
different frequencies and tem perature ranges.
5.2 PR ED IC TIO N S O F TH E AD W P M O D EL
The tim e (t) dependent polarizability, a (t) o f a dielectric m aterial is given by:
a ( t ) = a ( 0 ) - F ( t)
(5.1)
69
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
If we assum e a Debye m odel o f dielectric response, then the function F(t) is exponential,
given as:
F(t) = e x p ( - t / t )
(5.2)
where t is the relaxation tim e. The polarizability in the frequency dom ain, a ( o ) can be
obtained by taking a F ourier transform o f Eqn. (5.1).
, )X= -—
a (°)
a (co
:—
( 5 .3IX
)
1 + icot
M icroscopically, if one considers a charge jum ping betw een two states o f a potential well
as in fig. 5.1 at a specific rate, the following statem ents hold: at frequencies much lower
than the jum ping rate, equilibrium can be established m uch faster than the applied field,
and hence the polarization will be in phase w ith the applied field. W ith increasing
frequency, how ever, the polarization will no longer be able to keep in pace with the
quickly changing field, and hence, a(co) will start to gradually decrease. This decrease
will never be faster than the inverse frequency, i.e.: proportional to the rate o f change o f
field. If non-interacting, isolated pair o f states are considered, then the total polarization
can be obtained by sum m ing up the losses from all pairs [126],
a(co) =
e:R :
12kT
A j
sech'
U kT
1
(5.4)
1 + icox
Because the conductivity, cr, is proportional to co tim es the polarization, the m agnitude o f
conductivity m ust be a non-decreasing function o f co. In other w ords,
Re[a(o))] = Re[icoa(co)]
ct( co)
co
=
(5.5)
e: R :
A 'j
cot
s e c h ' -------12kT
V 2 k T j l + (cox)'
(0 .6)
70
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Incorporating a distribution function <|>(A,V) for the asym m etry energy and the barrier
height, and sum m ing up contributions from all A D W P configurations, the total ADW P
conductivity per unit volum e can be w ritten as:
dAdV
where X is the concentration o f A D W P configurations per unit volum e independent o f co
and T; R. is the spacing betw een the two w ells, typically ~ 4-5 A; e is the electronic
charge; Vm and Am are the cut-off energies for V and A respectively; and kT has the usual
m eaning. Lu et. al [51] have incorporated a sm all linear increase in the m axim um
asym m etry energy, Am w ith tem perature, i.e. Am=A0+'/T (Ao being the energy split o f the
two lowest states and y is a sm all constant). It is not unreasonable to do so because
therm al energy can cause a small change in structure that determ ines Am. In order to
calculate the effective relaxation tim e, t in Eqn. (5.7), w e consider the tem perature
regim e in w hich the dom inant charge transfer is by hopping over the barrier [127]. Then r
is given by:
i
exp[(V -!-—) / kT] - 1
e x p [(V ----- ) / kT] - 1
2
2
vk
( V )
T = To exp —
J
+----------------------
A
-sech ------
)
(5.8)
(5.9)
Here W [2 and W 2i are the transition rates from position 1 to 2 and from 2 to 1
respectively, and t 0 is a constant proportional to reciprocal o f phonon frequency.
71
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
D ifferent suggestions for the distribution function <t)(A,V) exist in the literature
[123,128,129], but a separation o f the form
ip( A, V) = p(A) ■g(V )
w ith
|J<j>(A, V)dAdV = 1
(5.10)
is com m only used. On general grounds, it can be argued that the distribution function,
p(A) m ust be sym m etric, since neither the right nor the left-hand well in fig. 5.1 is
preferred in the absence o f an electric field. Also to carry the calculation further, an
explicit form m ust be assum ed for the distribution function g(V). A ccording to Gilroy et.
al [123], who assum ed a reduction o f high energy barriers, the distribution function
(p(A,V) is w ritten as:
/
d>(A.V) =
sec h
V
(5.11)
where V0 is the w idth o f the distribution function. Other form s o f distribution function
have been used, but the integral in Eqn. (5.7) is such that the result is not very sensitive to
the exact form o f <j>(A,V); rather the w idth o f the distribution function, Vd, the maximum
cut-off energies, Vm and Am, and the relaxation time, t are the leading param eters that
determ ine the shape o f the conductivity curve.
5.2.1 A D W P C onductivity in 4K <T <150K
U nder the assum ption that V » k T and A » k T , the relaxation time, x in Eqn. (5.9)
reduces to
(5.12)
72
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
The second assum ption is less im portant; utm ost it can cause an error by a factor o f 2
[127], Also if w e assum e that V » V 0, then Eqn. (5.11) can be sim plified to:
(5.13)
i X
° T
v -
Substituting Eqns. (5.12) and (5.13) in (5.7), we get,
©t 0 exp
N e ; R ; co
,,f i L ' f
f
V]
dA e x p --------a(co,T ) = --------------- s e c h '
12kTA0V, j
U kTj
J \
V J
kT
1 + ( cot Q) ' exp
•dV
(5.14)
2V
kT
A llow ing v = cot0 exp(V7kT) and a = k T /V 0, the above integral reduces to the form:
«p<
N e : R : o)kT /
w
,
a(co.T ) = — :— —-------(“ T , ) tanh
2kT
6 A 0V o
rkrf )
dy
(5.15)
Since Vm» k T , the upper lim it can be taken as ac. The integral can be evaluated
num erically to be approxim ately equal to z / 2 for 0 < a< 0 .3 .
Finally. Eqn. (5.15) can be
expressed as:
, ” A_
N ;te2R : (okT
c t ( ( o , T ) « --------------------------( ® * 0 )
12A0V0
' ta n h
(5.16)
2kT
(5.17)
o(co,T) = C 'N Tco1**1 tanh
,2 k T ,
w here C'=7te2R2kTQa/l2A oV0. In other w ords,
Pt i
a (cj,T )acco pT
(5.18)
73
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
where [3 = l-a , and typically 0 < a< 0 .3 . Also Ttanh(Am/2kT) varies as Tn, w ith typical
values 0<r|<0.3 for 4K.<T<150K. [130]. Thus the A D W P m odel predicts an alm ost linear
dependence with frequency and a w eak therm al dependence at low tem peratures.
The .ADWP form alism was able to describe the experim ental conductivity o f
sev eral alkali gcrm anate and alkali tlurozirconate glasses at low frequencies (Hz-kHz)
and low tem peratures (4K.-150K.). The optim um values for different fitting param eters
were obtained as Vm= l0 0 0 K. V o=730 K, A«=5 1C, t o= l x l 0 " - 10"° sec, and N = 10‘ 10' ‘ cnv [130],
5.3 IN F L U E N C E O F F IT T IN G P A R A M E T E R S
The m ost im portant fitting param eters in Eqn. (5.7) are the Am, the m axim um
energy value for the asym m etry; Vm, the m axim um energy value for the barrier height;
the pre-exponential factor in relaxation; and N. the .ADWP concentrations in a unit
volume. The effect o f each param eter on .ADWP conductivity
(O a d w p)
w ill be briefly
discussed in this section.
5.3,1 V a ria tio n o f Vm
The variation o f A D W P conductivity w ith frequency for various Vm values is
show n in fig. 5.2 at room tem perature. It can be seen that Vm does not affect the
frequency and/or tem perature dependence o f conductivity as long as V m is greater than
6000K. In fig. 5.2, we have plotted the conductivity for different V ms and find that only
for V m > 6000K, th e trend o f conductivity does not change anym ore.
74
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
5.3.2 V ariation o f Am
It can be seen from Eqns. (5.17) and (5.18) that the m axim um asym m etry energy,
Am gives rise to the tem perature dependence o f
cta d w p -
W e assum e Am to have a small
linear increase w ith tem perature, such that Am=Ao-^/T, w here Aq=5K and y is a small
constant. As a consequence, an increase in Ao, (hence Am) results in a increase in the
tem perature dependence (exponent r\ in Eqn. (5.18)) o f a ADwp as shown in fig. 5.3. For
an increase in Ao from IK to 25K, the slope increases from ~ 0 .l I to 0.7 at 109 Hz. The
values o f the other fitting param eters used in the sim ulation are Vm=6000K, x0= 2xlO 'Ij
sec, y=0.05.
5.3.3 V ariation o f tq
The variation o f conductivity w ith frequency and tem perature is shown for
various x0 values in figs. 5.4a and 5.4b. From fig. 5.4a, we can see for xo= 5 x l0 '11 sec, the
conductivity increases linearly w ith frequency before becom ing invariant at frequencies
greater than ~109 Hz. As t 0 is reduced to sm aller values, the onset o f plateau shifts to
higher frequencies. The variation o f
c Ad w p
w ith tem perature (fig. 5.4b) indicates a peak,
w hich shifts to low er tem peratures as x0 is reduced. Therefore, the effect o f x0 is to shift
the peak along the tem perature axis or the plateau along the frequency axis; a sm aller x0
w ould bring the peak to a lower tem perature for any fixed frequency, or shift the plateau
to higher frequency for any fixed tem perature.
75
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
5.3.4 V a ria tio n of y
The small constant, y incorporated to describe the tem perature dependence o f
asym m etry energy, i.e.Am=A0-^/,T, usually lies in the range o f 0.02 - 0.08 in m ost oxide
glasses [130], Clearly, y determ ines the slope o f the curve betw een A and T, indicating
stronger tem perature dependence o f conductivity as y is increased as show n in fig. 5.5. In
fig. 5.5, we have plotted
Cta d w p
versus tem perature at 10', 10s and 109 Hz for two values
o f y=0.05 and 0.1 respectively, w hich show s the increase in the slope for higher y value.
5.3.5 V a ria tio n of V 0
The param eter V0 in the distribution function o f g (V )= l/V 0Sech(V /V 0) refers to
the width o f the distribution o f barrier energy. Vu determ ines the rate at which the
distribution o f barrier height g(V) decreases w ith energy. As show n in fig. 5.6, the rate o f
decrease o f g(V) is m ore rapid at lower energies for Vo- 5 0 K than for Vo=150 K. This
rapid fall in g(V) for sm all V0 values is reflected in the form o f a stronger frequency
dependence o f a,\Dwp.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
E
q
Figure 5.1: Schem atics o f asymm etric double w ell potential. V and A are the barrier
height and
asym m etry energy respectively.
The x-coordinate
‘q ’ is defined as
configuration coordinate while ‘E ’ is the energy coordinate.
77
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1 0 '1
r
ic r2 r
w
1 0 '3
r
c
0
1 0 ‘4
1 0 '5
r
r
1045
-
1 0 '7
r
'\ 0 ' a
r
ri
03
■*3^
—«
'•4— *
O
c
o
O
■O
cn
o
3
■a
O
ra 1 0 '11 r
13 1 0 ’12 r
V
o
8
-
O
r
V
0
o
□
*
0
E
1 0 ‘13 r
10 "
' ■■■■■■■
l u u iL
104
105
106
■'■■■■*
107
V =1000K
V =3000K
Vm=4000K
V =6000K
Vm=10000K
■ ■ * *“ “ t
10
* -4.
8
lllL
10s
Frequency (Hz)
Figure 5.2: V ariation o f AD W P conductivity at RT w ith frequency for different Vm
values. The values o f other param eters are Vo=730 K, Ao=5 K and t o= 5 x 1 0 '11 sec.
78
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
100 r
C/3
- o —o
■4— *
'c
jd
>
’>
O
13
■a
.- A '
•4—*
10
A_."
c
. O ''
O
*a
0
jg
13
..- A '
..-O ’ .-E3
''.
..-O ' - C l '"
.-•a'
Q ' " '
A 0=25K
o
A 0=5K
A
A 0=2K
O
no
E
C/3
□
<1
o
.A - '
100
10
1000
T em perature (K)
Figure 5.3: Tem perature dependence o f sim ulated A D W P conductivity at 109 Hz for
different asym m etry energy values, Ao- The sym bols represent the sim ulated data points
w hile the dashed lines are fits to pow er law equation o f the form Tr. T he values o f other
fitting param eters are Vm=6000 K and xo= 2 x l0 '13 sec.
79
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
2
0
•2
-4
O)
o
=2x10 J s
-o
=5x10 '2 s
=1x10',2s
=5x10 " s
■a
4
6
8
10
12
14
log (f, Hz)
(b)
10'5 r
'c
-Q
ra
10^ r
10'7
10
100
Temperature (K)
Figure 5.4: V ariation o f conductivity w ith (a) frequency and (b) tem perature for various
x0 values.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
10 r
A
10"
A
A
109 Hz £
4
A
A
A
A*
A
A
0°
O
3 , 10J r
&
O
108 Hz 3
>
0
O
••
8
O
3
■a
□ oa
c
10
:
10"
io
7H z
B
B
100
10
Temperature (K)
Figure 5.5: D ependence o f AD W P conductivity on y at different frequencies. The solid
sym bols are sim ulated points fory=0.05, w hile the open sym bols are for y=0.1.
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R e p r o d u c e d with p e r m i s s io n of t h e cop y rig h t o w n e r. F u r th e r r e p ro d u c tio n prohibited w ith o u t p e rm is s io n .
0 021s
- Vl)=50 K.
o Vo= 1 50 K
: ocsi
G 002
°« oo
0
100
200
V(K)
300
400
500
Figure 5.6: D istribution function g(V )= l/V 0 sech(V /V 0) as a function o f barrier energy,
V for tw o different V0 = 50 K and 150 K. values. g(V ) decreases rapidly at low er energies
for Vo=50 K than for 150 K.
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R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CHAPTER 6 EXPERIMENTAL RESULTS AND DATA ANALYSIS
The ionic conductivity, a , o f tw o m odel glass system s, nam ely lithium silicate
and potassium germ anate series has been studied as a function o f frequency and
tem perature. The experim ental results o f frequency and tem perature dependencies o f a
arc presented in this chapter individually for the two glass system s.
6.1 L IT H IU M S IL IC A T E G L A SS
6.1.1 Low F re q u e n c y C o n d u c tiv ity (H z-kH z)
The low frequency ionic conductivity o f a lithium silicate glass o f com position
35Li: 0 - 3 A l;0 3T P ; 0 5-6 lS i0 ; is m easured in the frequency range o f 'H z to kHz at
tem peratures from 4K to 300K.. The frequency dependency o f cr is show n in a log-log
plot for tem peratures below room tem perature (RT) in fig. 6.1. At low tem peratures
(<150K ), the conductivity increases linearly with frequency with a slope alm ost close to
unity. The conductivity in this region has been established to be due to therm ally
activated
fluctuations
of
ADW P
configurations
[51,130,131],
w hich
can
be
approxim ately expressed as:
ct( o ,T )
= C T 7cop
(6.1)
where, C is a constant independent o f frequency and tem perature but dependent on glass
com position; (3 = 1.05 for this com position. As tem perature increases above -1 5 0 K . the
conductivity plots start to bend up at the low frequency end and finally show a plateau at
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- 298K. This frequency independent conductivity represents the dc conductivity, e rr a n d
has a value o f about 3 x l0 '9 Q 'c m '1 at 298K. for this glass com position. The slope o f a
versus f curve around 104 Hz is no longer near unity, but reduces to a lower value o f
about -0 .6 . Thus at room tem perature (300K.), the low -frequency conductivity follows
the UDR, described em pirically by Eqn. (6.2), w here conduction due to hopping o f a
single alkali ion becom es dom inant.
o ( gj,T )
= a (0 ) + A'co5
(6.2)
where, a (0 ) is the dc conductivity, A' is an em pirical tem perature-dependent constant and
s=0.6.
The tem perature dependence o f ionic conductivity for several frequencies is
shown in fig. 6.2. One can clearly see that conductivity increases with a weak
tem perature dependence at tem peratures <150-200fC, and increases m uch more rapidly
for tem peratures >200K. This w eak dependence for T < 150K is characteristic o f the low
tem perature ADW P fluctuations, where only the localized m otion o f the jellyfish
structures is im portant. At T>200K, the charge carriers, m ainly the m obile alkali ions
acquire enough therm al energy to hop over the long range diffusion barrier, resulting in a
rapid rise o f conductivity.
A closer exam ination o f the tem perature dependence at low T (<150K) show s that
the data could be fitted best by an em pirical relation o f the form,
ct ( c j ,
T) = B’(co) e x p (a T )
(6.3)
where B'(co) is the pre-exponential factor and a is the em pirically determ ined fitting
param eter. This em pirical fit agrees w ell w ith the observed tem perature dependence o f
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
experim ental data for several f values in the range o f 10 Hz to 103 Hz, show n as dashed
lines in fig. 6.3. W e have tried other relationships like a a T ' and ctT a exp(-E/kT), but
none describe the data as well as Eqn. (6.3) does in this experim ental range. T he value o f
a is found to be around '4 x 1 0 ° BC1 and varies only slightly with frequency.
6.1.2 M icro w av e C o n d u ctiv ity (50 M H z - 3 G H z)
The frequency dependence o f <r betw een 50 M Hz and 3 GHz for tem peratures
from RT (300K ) to 400K. is shown for the lithium silicate glass in fig. 6.4. At m icrowave
frequencies, once again the conductivity shows a nearly linear increase w ith frequency
resem bling the low f- low T region. The slope o f log c vs. log f plot is close to 0.9 at the
low er end o f the frequency range (<4xlO s Hz), gradually increasing to l .l at the higher
end (> 2 x l0 9 Hz) o f the M W frequency region. Thus, as can be extracted from figs. 6.1
and 6.4. the com bination o f low and high frequency conductivity at RT show s that as
frequency increases, the slope o f a increases from s=0 to s~0.6 to s > l.
The variation o f conductivity w ith frequency betw een 10s Hz and 2 x l0 9 Hz at
300K is analyzed by using an em pirical fitting function o f the form a(co)= Bto15 (Eqn.
(6.1)), w here B=CTy, <a=27tf and P = l. It w as found that this function fits the data well in
a narrow range betw een 4 x l0 8 Hz and Ix lO 9 Hz, but show s discrepancy at the low er and
higher frequency end. This indicates that
ct
is an increasing function o f frequency.
T herefore, the sim plified form o f the A D W P conductivity, as given by Eqn. (6.1), is not
sufficient to m odel the frequency dependence o f m icrow ave conductivity at room
tem perature and above.
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The variation o f a w ith tem perature in the m icrow ave region show s a very w eak
dependence as seen in fig. 6.5. Conductivity increases slow ly as the tem perature is
increased from RT to about 400K . The data are fitted to an em pirical relation sim ilar to
Eqn. (6.3). This em pirical fit is show n as dashed lines through the data points in fig. 6.5.
Ln this high f - high T region, the value o f a is a v ery sm all num ber, typically lying in the
range o f 3x10'"' K.'1 - 1.6x10''’ K.'1. The variations o f the em pirical constants, nam ely
a and B \ w ith frequency for the entire range betw een 10 Hz and 1010 Hz are plotted in
fig. 6.6(a and b). B' increases as a pow er law o f frequency w ith an exponent close to 1.1,
i.e. ct - o ' 1 (see fig. 6.6(b)). The a values for different frequencies are generally small
and lie w ithin 0.0016 to 0.004 (see fig. 6.6(a)). T hese can be considered nearly constant
in view o f the large frequency range, although its value at high frequencies is consistently
sm aller than at low frequencies. Because o f the low value o f a , one can say that data fit
to Eqn. (6.3) is approxim ately the sam e as w ith Eqn. (6.1) for T<150K..
6.1.2.1 Data A nalysis using A D W P m odel
Empirical Analysis
Prelim inary analysis o f the low T- low f and high T- high f d ata using the
em pirical relation given by Eqn. (6.3) was done to establish w hether the conductivity in
the two regions m ay have a com m on origin. Em pirical extrapolation o f the low T- low f
data to RT at different frequencies is done first, and the extrapolated RT conductivity
then com pared w ith the experim ental high T - high f conductivity. Fig. 6 .7 a shows that
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this extrapolation seem s to connect well w ith the high frequency experim ental data
points. The dashed line in the plot has a slope o f 1.03. A sim ilar extrapolation was done
to study the connectivity o f cr in the two regions at tem peratures above 300K. for which
m icrow ave data exist. The outcom e o f such an analysis yielded results sim ilar to that for
300K, with the slopes (for different tem peratures) ranging betw een 1.01-1.04. This
suggests that the m echanism o f conduction in the two regions m ay indeed have a
com m on origin [132].
Theoretical Analysis
The data w ere also analyzed theoretically using the A D W P form alism , Eqn. (5.7).
The low tem perature a data betw een 4K and 100K, for w hich the ADW P m odel is
established, is extrapolated to room tem perature (300K) at different frequencies using
Eqn. (5.7). The values o f the fitting param eters used in this extrapolation are same as
those at low T - low f work o f Lu and Jain [133], viz. R=4 A. Vm= l0 0 0 K , Vo=730K..
to ^ lx lO '0 sec. and A,=5K. T he extrapolation points are show n in fig. 6.7b as solid circles
for three frequencies (betw een Hz - kHz) together with the high frequency data points.
Due to the com plicated m athem atical sim ulation involved in solving Eqn. (5.7), the low
frequency extrapolation in fig. 6.7b w as done only for three frequency points, viz. 5 kHz,
10 kHz and 20 kH z, so as to em phasize the trend. It can be seen that the extrapolated
results do not m atch up w ith the high frequency experim ental data. Thus, the results o f
this prelim inary w ork on lithium silicate glass show that the analysis o f the data
according to the AD W P m odel does not exactly relate to the linear behavior at RT - high
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R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
f. This discrepancy in the extrapolation im plies the need for an alternate fitting form,
either through the com plete expression o f the A D W P m odel w ith fining param eters that
are significantly different from those values at low T and low f, or an alternative
em pirical equation, capable o f predicting the observed f and T dependence o f the
observed o\iw .
In order to verify w hether the low T - low f A D W P param eters are at all capable
o f predicting ctmw, we have attem pted to m odel the RT high frequency data using the low
T - low f param eters.
C om puter sim ulation o f the A D W P conductivity,
ct a d w p
using
Eqn. (5.7), as a function o f tem perature is show n in fig. 6.8a for different frequencies
using a com m only used distribution function o f the form <j>(A,V)=(l/AaV0) scch(V/V„)
[134], From fig. 6.Sa, w e note that the conductivity has a m axim um around 120K for 10'
Hz. which shifts to higher tem peratures as f increases and finally m oves out o f the
tem perature range at >10;: Hz. T he term a u /f l^ c o V ) in Eqn. (5.7) is the source o f
m axim um , w hich occurs when the condition co t« 1 is satisfied. Also as f increases, the
predicted
frequency dependence o f sim ulated
ctadw p
decreases
from unity before
becom ing invariant at frequencies greater than 106 Hz (see fig. 6.8b). Since this trend is
not observed experim entally, the same param eters used at low T and low f cannot be
valid at m icrow ave frequencies. Incidentally, we have also sim ulated a(co,T) using a
different distribution function <t>(A,V)=(l/AaV m) [135] and found qualitatively sim ilar
results, indicating that cr(co,T) is not very sensitive to the exact form o f <p(A,V); instead
the cu t-o ff energies, Vm and Am, the relaxation tim e t , and the w idth o f the distribution
function are the leading param eters that determ ine the shape o f the conductivity curve.
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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Since the effect o f t u is to shift the peak along the tem perature axis o r the plateau
along the frequency axis, a sm aller t u w ould bring the peak to a low er tem perature for
any fixed frequency, or shift the plateau to higher frequency for any fixed tem perature.
Also invoking the condition cot0=1, we m ay use its value as low as t o= 2 x 1 0 '1j sec. Such a
xu is justified by nuclear spin relaxation (NSR) m easurem ents on som e fluorozirconate
glasses at low tem peratures (<100K) [136], w here instead o f a localized single ion
hopping process, an excitation o f .ADWP configurations form ed by several atoms is
observed. The .ADWP sim ulation predicts that the variation o f cr with frequency is linear
in the m icrow ave region as observed experim entally, while the plateau predicted by the
m odel now occurs at frequencies >5xiO n Hz. The sim ulated
c j.\d w p
using the new set o f
param eters is show n as a function o f tem perature for various frequencies in tig. 6.9a; fig.
6.9b in this case show ing the frequency dependence at RT. The values o f ADW P
param eters used in the sim ulation are Vm=6000K . Vo=3000K . Ao=5K, M ^ x l O 1* cm J and
To-ZxlO '1"' sec, w hich are significantly different from those at low T and low f.
6,1.3 F a r In fr a re d C o n d u c tiv ity (>0.5 T H z)
The far infrared (FIR) conductivity,
ctfir
is obtained at frequencies > 5 x l0 ‘‘ Hz at
RT for the lithium silicate glass and is show n in fig. 6.10. T he conductivity obtained for
frequencies < 5 x l0 M Hz has been calculated from the K ram ers-K ronig extrapolation o f
the reflectance data. A t the low frequency flank o f the spectrum ,
cjfir
varies as - o ' ,
before giving rise to absorption peaks due to localized vibrations o f alkali ions and other
structural units at a few T H z and higher. The shoulder around 5 x l0 i: Hz (~160 c m 1,
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lcrrT = 30 GHz), indicated by an arrow, corresponds to the vibration o f a single Li~ ion
in its localized site in the glass, which agrees w ith the published results on the lithium
disilicate glass by Kapoutsis et al. [137]. The asym m etry o f this band suggests the
existence o f m ore than one band. The peaks at frequencies - 1 0 1' Hz and above are
attributed to the rocking m otion, bending m ode and asym m etric stretching o f Si-O-Si
b ad g es [137],
6.1.4 S u m m a ry
The variation o f conductivity over a broad range o f frequency, from 10 Hz to 10‘J
Hz at different tem peratures is shown for the lithium silicate glass o f com position
3 5 L i;0 -3 A l:0 ;-1P ;O f6 1 S iO ; in fig. 6.11. Region I represents the frequency independent
dc conductivity. cr0. The tem perature dependence o f a„ is A rrhenius with dc activation
energy £^-=0.67 eV. Region II represents the pow er law region with slope s-0 .5 - 0.7,
typically observed at low frequencies (-H z - kHz) and interm ediate tem peratures
(200K <T<500K ). This is called the GDR or Jonscher-N gai region. The tem perature
dependence is also .Arrhenius w ith activation energy Eac « (l-s )-E jc. Region Ilia is the
nearly constant loss region (NCL) with slope s~1.0 - 1.15 observed at low frequencies
and low tem peratures (4K <T<100K ). The conductivity show s w eak thermal activation in
this regim e. R egion IHb is the m icrow ave region (100 M Hz - few GHz) where the
conductivity has slope ~ 1.0, sim ilar to region Hla. T em perature dependence is again
w eak in this region. Finally, region IV represents the far infrared region. Conductivity
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varies as —co2 before giving rise to resonant peaks at very high frequencies (> 1012 Hz).
C onductivity in this region is tem perature independent.
6.2 PO TA SSIUM G E R M A N A T E G LASS SER IES
The
dispersion
o f ionic
conductivity
in
potassium
germ anate
series
of
com position, .tBCaO (l-.t)G eO :, 0.0023< .t <0.247, was studied at different frequencies
and tem peratures. T he ionic conductivity results are presented for two BCO content (24.7
mol% and 7.4 m ol% ) glasses for sim plicity. The com position dependence o f a is studied
and correlated to the structure o f the glass netw ork. The dependence o f the ADW P
param eters on the glass com position in term s o f the structural param eters is also
presented.
The chem ical and physical structures o f this glass series are w ell know n by XPS,
X -ray and neutron scattering, [R, Raman, EXAFS etc [138,139,140,141,142], The
germ anates, in general, show either a m inim um or a m axim um in their physical
properties as the alkali concentration increases. This anom alous behavior w as explained
as due to the conversion o f GeC>4 tetrahedral to GeO& octahedral units w ith the addition o f
alkali oxide up to 20-25 m ol% . Higher alkali concentration causes the form ation o f non­
bridging oxygen (N B O ) containing germ anate tetrahedra.
6.2.1 Low Frequency C onductivity (H z-kH z)
The low frequency conductivity betw een 10 Hz and 100 kH z for the glass
com positions x= 0.0023, 0.02, 0.074 and 0.247 w as m easured at tem peratures betw een 4K
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R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
and 350K. The frequency dependence o f a for the 24.7 m ol% K 2O glass is show n in fig.
6.12 for different tem peratures in a log-log plot. T he low tem perature data are from the
w ork o f Lu [143], Like in lithium silicate glass, a increases linearly with frequency, the
slope being close to unity for tem peratures below 200K. This linear frequency behavior
can be described by Eqn. (6.1). In fact, the linear A D W P region is observed at low
tem peratures in all com positions w ithin this glass series. For tem peratures i-LZOOK., the
increase in cr is no longer linear and contribution from diffusive single ion hopping
begins to appear at the low frequency end o f the plot. T he frequency independent dc
conductivity (ctjc) region appears at T > 346.3K. It increases w ith tem perature and finally
at 394.7K, it spreads over at least two decades o f frequency. At the higher frequency end,
>10J Hz, <7 dc com ponent is small. The transition from dc to ac region is rather smooth,
with the slope increasing from 0 to 0.6 corresponding to the dc plateau and the L D R (i.e.
the sublinear pow er law) regions, respectively.
6.2.1.1 Scaling properties o f conductivity spectra
The ion transport in glasses has been thoroughly studied by using the conductivity
(a ) and dielectric m odulus (M *) spectra as a function o f tem perature, frequency and
com position. Despite the vast am ount o f experim ental results, there is generally no
accepted view about the m ost appropriate m ethod o f analysis. R esults o f analysis using
the M* form alism and cr have been contradictory w ith respect to a very im portant aspect:
the influence o f com position, in particular the influence o f density o f m obile ions, on the
ion transport properties. It is observed that M" (=Im .M *) spectra becom e narrow er as the
92
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ion concentration decreases. This effect is understood to be due to changes in the ion
relaxational characteristics w ith concentration. O n the other hand, the shape o f the real
part o f cr spectra does not change significantly w ith concentration o f m obile ions. Kahnt
perform ed conductivity m easurem ents on sodium silicate and sodium germ anate glasses
o f varying N aiO concentration [144]. H e found no influence o f alkali ion concentration
on the shape o f conductivity spectra, even for glasses w ith alkali content below 1 mol%.
In other w ords, the frequency dependence o f conductivity spectra can be collapsed onto a
single ‘m aster’ curve reflecting that the transport m echanism is quite independent o f
m obile ion concentration. Low frequency m aster curves have also been obtained for
glassy electrolytes such as 0.3A gI • 0.7AgPO3 [145] over a frequency range o f 10 Hz to
10 VlHz at different tem peratures. M ost rem arkably, frequency-dependent conductivities
o f other solid electrolytes follow alm ost the same m aster curve. This was recently shown
for quite a num ber o f crystalline and glassy system s [146,147],
W e note another recent paper b y Roling et al. [147] reporting that the frequency
dependence o f conductivity o f glasses can be reduced into one ‘m aste r’ curve irrespective
o f chem ical com position by using a scaling factor, only in the cross-over region from dc
to dispersive conduction spanning about two orders o f m agnitude the ‘m aster’ curve is
com position dependent. T hey contend that, in a first approxim ation, the UDR. is unrelated
to c o ^ in M" plot and its origin is independent o f m obile ion concentration. We have
conducted an analysis to verify if the conductivity data for the potassium germ anate glass
scries can also be fitted to a sim ilar ‘m aster’ curve. To do this w e have considered the
low f conductivity plots for three K-Ge glass com positions, nam ely 0.0023, 0.02 and 0.2,
93
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
and scaled by a factor o f ( o) x/T ctjc) suggested b y R oling et al., where x is the
concentration o f alkali oxide and co (=2rcf) is the angular frequency. The results in fig.
6.13 show that the scaled curves do not fall into one m aster curve; the discrepancy
prevails for m ore than three decades o f frequency beyond the dc plateau. In fact,
considering the sublinear com position and com plex tem perature dependence o f the
conductivity at low tem peratures [133], one expects for the present glass series that the
curves in fig. 6.13 will not m erge at the m uch higher values o f the scaling factor. In fig.
6.13 the curve for the low est potassium content glass is the highest and has the steepest
slope beyond the crossover region; converse is true for the highest potassium content
glass.
6.2.1.2 C oncept o f M ism atch and Relaxation
The possible existence o f one single low -frequency conductivity m aster curve,
w hich is exhibited by som e crystals, glasses and m olten electrolytes raises the question o f
w hy the long range dynam ics o f these otherw ise different system s be rem arkably similar.
The c o n c e p t o f m is m a tc h a n d r e la x a tio n (CM R) w as proposed by Funke et al. [145] to
explain this sim ilarity in the shape o f the conductivity curve. A ccording to them , each
elem ental hop o f the ion is supposed to create som e m ism atch betw een the ion and its
neighborhood. The m ism atch m ay be interpreted as the distance betw een the actual
position o f the ion and the position at w hich the ion w ould be optim ally relaxed with
respect to the arrangem ent o f its neighbors. As tim e proceeds, the system finds a w ay to
relax the original m ism atch. Tw o particular w ays o f m ism atch relaxation can be
94
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
visualized - (1) the single-particle route, where the ion itse lf perform s a correlated
backw ard hop, and (2) m ulti-particle route, where a rearrangem ent o f neighboring ions
takes place. W hich w ay the relaxation proceeds depends on the tim e constants or
relaxation rates o f the two processes.
The central hypothesis o f the C M R is that the rates o f relaxation via the singleand m ulti-particle routes are coupled to each other in a synchronous fashion. In other
words, the tendency o f the central ion to hop backw ards is proportional to the tendency o f
the surroundings to rearrange. This means the particular shape o f conductivity dispersion
is well defined. The m odel m aster curve is defined by the function given by
onset
(6.4)
where. E ,'1 denotes the inverse function o f the exponential integral.
E, (x) = I-— dv
, y
(6.5)
In Eqn. (6.4), corset is the onset frequency o f dispersion w here <j(coonSet)= l-303-a(0).
Apart from ct(0) and
coonSe t,
which are needed for norm alization, there are no adjustable
param eters. The shape o f Eqn. (6.4) shows that the slope o f conductivity curve increases
continuously, but never to exceed unity. Therefore, the concept o f a single pow er law
(UDR and/or NCL) has to be abandoned. In other w ords, the pow er law s (Eqns. (6.1) and
(6.2)) can be used to describe the spectra only w ithin very narrow frequency ranges. A
direct com parison o f the m aster curve (Eqn. (6.4)) and experim ental data has been
provided for R b A g Jj and a few o th er crystals by Funke and W ilm er [148],
95
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
An im portant im plication o f the CM R m odel concerns the frequently observed
nearly linear frequency dependence o f conductivity (i.e. ~ col,Q). A ccording to the model,
the N C L behavior is sim ply an extension o f U D R that is due to single-ion hopping i.e. the
two universalities, nam ely, universal dynam ic response (U D R) and the nearly constant
loss (N CL) b ehavior have one com m on origin. In order to test the validity and
applicability o f the C M R model, w e have perform ed a sim ilar scaling analysis for 1%
G d'*-doped C eO : ceram ic, 11% G d ^ -d o p ed highly defective CeCS polycrystal, and
x K ;0 ( l-x )G e O : glass series. The result o f the analysis for the 1% doped crystal is shown
in tig. 6.14a in the form o f a ‘m aster’ plot. T he conductivity and frequency scales are
norm alized w ith respect to dc conductivity and coonsct respectively. T he solid line is the
CM R m aster curve described by Eqn. (6.4) w hile the sym bols are the experim ental low
frequency data. From fig. 6.14a, one can clearly see that there is discrepancy between the
experim ental and m aster curve, particularly beyond the onset o f dispersion. Results o f
scaling analysis on the 11% doped sam ple also show a sim ilar discrepancy betw een the
experim ental data and the C M R function (see figure 6.14b).
A nalysis on the binary germ anate glass com positions also show s sim ilar results,
with m ism atch betw een the m aster curve and experim ental d ata in the dispersion regime.
Scaling plots are show n for 24.7 m ol% KiO glass at two different tem peratures in fig.
6.15. In addition, here we have included the experim ental conductivity data at m icrowave
frequencies for these tem peratures. C learly the m aster curve overestim ates the observed
ctmw
in both cases, b y alm ost an order o f m agnitude, thereby show ing disagreem ent with
the C M R m odel. V ery recently, Funke has m odified his m odel to correct for this
96
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
discrepancy. He has suggested the inclusion o f a second term in Eqn. (6.4), to account for
the structural properties o f the environm ent.
6.2.2 M icrow ave C onductivity (100 M H z - 3 G H z)
M icrowave conductivity o f the K-Ge glass series was m easured betw een 100
M Hz and 3 GH z at RT and above. The variation o f
g .m w
w ith frequency for the glass
com positions 0 .2 4 7 ^ 0 - 0 .753GeC>2 and 0.074K.20-0.926Ge02 is show n in figs. 6.16 and
6.17 respectively for tem peratures from RT (300K) to 400K . From both plots one can see
that a increases with slope o f unity for a narrow range betw een '3 x 1 0 ^ Hz and 109 Hz.
In order to establish that cr in the tw o regions, nam ely, low T - low f and high T - high f
have the sam e origin, the data in the two regions have been analyzed using the em pirical
relation o f the form sim ilar to Eqn. (6.1). The low T (4K. - 100K.) and low f (10 Hz - 100
kHz) data are extrapolated to room tem perature using Eqn. (6.1) and com pared with the
high frequency experim ental data. The result o f this em pirical extrapolation is shown in
figs. 6.18a and 6.18b for 24.7 m ol% and 7.4 m ol% potassium content glass respectively.
O ne can clearly see that the tw o regions connect reasonably w ell, indicating the
m echanism o f conduction to be related in the two regions, as show n before for a silicate
glass in fig 6.7a. The dashed line join in g the sym bols is a least square fit with slope - I . I.
Em pirical analysis perform ed on other glass com positions yielded results sim ilar to that
presented for these two glasses.
The tem perature dependence o f
gmw
follow s a T ' pow er law, w ith y values
consistently being higher than that observed at low T and low f b y Lu et al. [133] for this
97
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
glass system . The values o f y for the all glass com positions at low T - low f and high T high f are given in Table 6.1 for the sake o f com parison. T he high values o f y at MW
frequencies suggest that the localized fluctuations o f A D W P structures increase with
tem perature due to the increase in asym m etry energy betw een the two sites o f the
potential well. The tem perature dependence o f
ctmw
for 24.7 m ol% alkali content glass is
show n in fig. 6.19 for several frequencies. The dashed lines are fit to T y pow er law. The
variation o f y w ith alkali concentration in the two regim es is com pared in figs. 6.20a and
6.20b for the K.-Ge glass series. In the low T - low f regim e, the tem perature exponent
decreases first rapidly as alkali concentration increases and then show s m uch less change
for alkali concentration > 2 mol% (fig. 6.20a). T his decrease in y is explained in terms o f
the
decrease
in
strain
energy
for
AD W P
configurations
due
to
progressive
depolym erization o f the glass structure upon the addition o f alkali ion [133]. The
com position dependence o f y in high T - high f region (fig. 6.20b) also shows an initial
decrease as concentration increases from 0.23 to 2 m ol% . A bove 2 m ol% , y increases
slow ly until the concentration reaches - 2 0 mol% , after w hich it increases rapidly.
The variation o f the m agnitude o f o\iw w ith alkali concentration in the glass has
been also studied. T he results in fig. 6.21 show that Cmw
t
increases w eakly with alkali
concentration. For exam ple, at I GHz, conductivity increases b y only a factor o f 2 upon
increasing the potassium content from 0.23 to 25 m ol% (factor o f 100). This is in stark
contrast w ith the dependence o f low T - low f conductivity, w here a increases by almost
an order o f m agnitude upon increasing the alkali content from 0.23 to 25 m ol% [143], Lu
et al. studied the variation o f ADW P configurations (A D W PC ) with glass composition
98
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
for a binary lithium germ anate series at low T and low f. It was observed that the ADW P
concentration follows a pow er law, i.e. N a .t°'79, w here .t is the lithium oxide content
[133] as show n in fig. 6.22. This sublinear dependence on com position suggests that
more than one alkali ion effectively contribute to one A D W P configuration, which also
includes the netw ork form er atoms.
6.2.2.1 Data A nalysis using ADW P M odel
Having established that the conduction m echanism at low f - low T and high f high T regions have essentially a com m on origin, the
ctmw
was analyzed using the
com plete .ADWP formalism to establish the origin o f conduction at m icrowave
frequencies in the present oxide glasses. Sim ulation using Eqn. (5.7) perform ed on the K.Ge glass series showed that most o f the AD W P param eter values rem ained about the
sam e as in the silicate glass, except for x0, w hich decreased from 2x10‘13 sec in the
silicate to 5 x l 0 'u sec in germanate glass. A list o f A D W P fitting param eters is show n in
Table 6.2 for both glass systems as a function o f glass com position. It can be seen that
Vm, V0 and
t0
do not change very m uch with alkali concentration in germ anate glass
scries, while the concentration o f A D W P configurations, N, varies with alkali content.
This im plies that the first three A D W P param eters are not sensitive to the details o f
structure o f the glass netw ork, w hich changes dram atically w ith alkali concentration for
this glass series. The change in Cmw
t
w ith alkali concentration m ostly occurs due to the
change in concentration o f ADW P configurations, w hich is sim ilar to w hat Lu et. al
observed at low T - low f regime. The com position dependence o f N observed at high T
99
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
and high f is plotted in fig. 6.23. The value o f N increases from 1 .8 x l0 20 cm 'J at the
lowest alkali concentration to about 3 .7 x l0 20 cm '3 at the highest alkali concentration. The
results are tabulated in Table 6.3 for all com positions in this glass series. O n com paring
the variation o f N w ith alkali content at low f - low T and high f - high T (figs. 6.22 and
6.23), we see that the increase o f N with alkali content is m uch w eaker in the m icrowave
frequency regim e. The variation o f N at m icrowave frequencies follows an approxim ate
pow er law, N a
.r being K ;0 concentration. M oreover, the value o f N obtained from
sim ulation at M W frequencies is higher by an order o f m agnitude in the silicate glass and
by alm ost two orders o f m agnitude in the germ anate glass series w hen com pared to the
value obtained at low tem peratures and low frequencies.
100
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Glass Series
K -G e (m oi% )
low f - low T [133]
Y
high f - high T
(present work)
Y
0.23
0.24 ± 0.03
0.675 ± 0 .0 5
2
0.08 ± 0.02
0.503 ± 0.03
5
—
0.661 ± 0 .0 5
7.4
0.08 ± 0.02
0.742 ± 0.07
10
-
0.643 r 0.07
15
—
0.725 ± 0.05
-
0.701 ± 0 .0 4
20
24.7
0.10 ± 0.01
0.861 ± 0 .0 3
Table 6.1: C om parison o f tem perature exponent, y for K -G e glass series at low T - low f
and high T - high f regions.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Sum m ary o f high frequency AD W P fitting param eters for K -G e glass series
Vm (K)
Ao(K)
N (cm 'J)
t 0 (sec)
0.23
6013
6
1.81x10-°
5 x 1 0 '“
2
6000
5
2.78 xlO :o
5
6000
5.7
1.94 xlO 20
7.4
5973
5
2.29 xlO 20
10
6007
6.3
2.62 x 102°
15
6001
5
3.21 xlO 20
20
5998
5
3.26 xlO 20
24.7
6000
j
C om position
m ol%
•y
”
i
j
i
!
'V
•*
3.65 xlO 20
Sum m ary o f high frequency ADW P fitting param eters for lithium silicate glass
Vm (K)
Ao(K)
N (cm 0 )
to (sec)
6000
5
3 x l 0 19
2x10 '13
Table 6.2: Sum m ary o f high frequency A D W P fitting param eters for the K -G e glass
series and lithium silicate glass.
102
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
i
Glass Com position
AD W P C on centration,
K20 (m ol% )
N (cm"3)
0.23
1 .8 1 x l0 20
2
2 .7 8 x l0 20
5
1.94.x 1020
7.4
2.29x1020
10
2.62.x 1020
15
3 .2 1 x l0 20
20
3.26x1020
24.7
3 .6 5 x l0 20
Table 6.3: C alculated ADW P concentration values in K -G e glass series at high
frequencies.
103
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
35L i20 -3 A120 3-1P20 5-61S i 0 2
1 0 -7
10^
_
E
_o
10'9
•f* 1 0 ,c
u
■o
c
o
O
7K
50K
10"
slope=1.Q
1 0 '2
150K
200K
O— 225K
250K
a - 275K
298K
10 ’3
10’
Frequency (Hz)
Figure 6.1: Low frequency ionic conductivity o f lithium silicate glass o f com position
3 5 L i2 0 o A l20 3 -lP 205-61Si02 at tem peratures from 7K to 300K. T he dotted line is
reference line with slope = 1.0. The uncertainity in the data is sm aller than the size o f
symbols.
104
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
100 Hz
500H z
1kHz
5kHz
10kHz
50kH z
35LLO 1 P00 K3ALO- 61SiO,
E
_o
CO
■>
o
Zj
~0
c
o
O
0
50
100
150
200
250
300
350
Temperature (K)
Figure 6.2: Tem perature dependence o f conductivity o f 35L i20-3A l203-lP :0s-61S i0
glass for frequencies betw een 100 Hz and 50 kHz.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
55L i'0-3A l--0'-lP '0> -61S i0-'
“1—:—i—i—|—!—i—!—r
10'!
...........A ...............
■A . A ......
-A -A 1 0 -10
-
-CCQ-C”' - ....
.1
-o-
o•-
.....
_u
>,
>
10 '
"
u
3
TrD
;
O
O
♦
i a ,:
*
-O --
..▼▼T-T"
..0 0 0 - 0 - o .... c -
10
•
-o
13
20
40
60
80
100
120
Temperature(K)
Figure 6.3: V ariation o f conductivity with tem perature in the region o f 4K < T < 100K.
for frequencies 50Hz, 100Hz, 500Hz, 1kHz. 5kHz, 10kHz and 20kH z from bottom to top
respectively. The dashed line is em pirical fit to Eqn. (6.3).
106
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
35%LLO 61% SiO, 1%P,CL 3%AL0.
300K
329.1K
345K
361.4K
378.7K
396.2K
•a
slope=1.0
108
Frequency (Hz)
Figure 6.4: M icrow ave conductivity o f 35L i20-3A l203-lP20s-61Si02 glass from RT to
400K. The dashed line is a reference with slope = 1.0.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
35Li20-3Al203-lP205*6lSi0a
E
.o
A—
C/D
A "'
A -
>>
•I
t5
U
■O
r;
o
O
10'4
280
300
320
340
360
380
400
420
Temperature(K)
Figure
6.5:
Tem perature
dependence
of
m icrow ave
conductivity,
ctmw
for
S S L iiO o A F O j-lP iO s ^ lS iO : at frequencies 100 M H z, 500 M H z, 1 G H z, 2 G H z and 3
GHz from bottom to top respectively. The dashed lines are fit to em pirical Eqn. (6.3).
108
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
10-1
(a)
35Li: 0-3 Al->03- lP : 0 5-61S iO ;
10 -2
I
••
••
a
•••
10 ‘3
i 11 m i j
1 0 '3
10-4
-
i
11I I K
i
iiui
i m m
i l i m a
I IIIIU
i
l i n n
iu L
I l i u m
I III
Ill
♦♦
(b)
10-5
1 0 '6
>
10-7
.o
CO
1 0 '8
CD
1 0 ‘9
10-1°
10-11
10-12
<►
11*1 nnJ
101-13
101
■ ' 111“ ^
102
■
1 03
ja L
i l
1 0s
10s
* ___ I I " i n r t ___ I m
1 04
i n
m ill
riiimi
107
i »»miJ
10a
i 11 iiml
109 1 0 1°
Frequency (Hz)
Figure 6.6: Frequency dependence o f em pirically determ ined constants, (a) a and (b)
for a w ide range betw een 10 Hz and 10l° Hz. a is nearly constant, w hile B' varies
- t o 1!. The dashed line is the least square fit w ith slope 1.1.
109
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
log(Frequency, Hz)
-2 -------:------ ;------ 1--------------------- r-3-
3 5 L i;0 -3 A l:0 r lP :0 ;-6 lS i0 ;
-4 r
5
-5
co
>
-6
>
'■3
-7
1o
%
o
-I
I
j
.«
-10
(b)
-11
j
-12
1
2
3
4
5
6
7
8
9
10
log(Frequency, Hz)
Figure 6.7: (a) C om parison o f em pirically extrapolated low f - low T conductivity data at
RT using Eqn. (6.3) and experim ental high f - high T data. Results show that this
extrapolation connects the data in the two regions reasonably well. The dashed line is a
least square fit w ith slope 1.03. (b) Extrapolation o flo w f - low T using the form alism o f
A D W P (Eqn. (5.7)). The values o f fitting param eters are as indicated in text. D ashed
lines are least square fits to the data.
110
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
X XX
cn
'c
o
-Q
0-r
CD
Q
<
C
100
10
1000
Temperature (K)
T=500K
-7
Q
<
JO
05
-8
O
-9
-10
2
3
4
5
6
7
8
9
10
iog(f, Hz)
Figure 6.8: (a) C om puter sim ulation o f AD W P conductivity using Eqn. (5.7) as a
function o f tem perature for several frequencies. The values o f fitting param eters are
Vm=1000 K. Vo=730 K, ^ = 1 x 1 0 ^ sec and 4 j =5 K. from low T - low f w ork o f Lu and
Jain [133].
(b )
Frequency dependence o f c t a d w p at 300K.
Ill
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
101
.
10 °
w
c
13
jQ
La
▼
10-1
10-2
-2-
10-3
a.
5
Q
<
c
■a
a)
▼
.
V
V
A
▲
A
A
A
10-4
■
1 0 '5
□
’ 1 0 11 H z
----------T
» »
v V V V 7 V V W 101° H z
E
10-7
W
1 0 ‘8
a
A * *
* A A A A 1Q9 H z
A
A
A
A
A
i-- l
1
A A
1 Q 8
^
107 H z
■ •
•
O O OO O O,
o
o
(a)
A
a □ □ □ □ o □ aa 1 q6 Hz
□
•
1 0 '9
A
■ ■ ■■ ■ ■
10-6
13
Hz
1 0 ^ -1 0
1
I
• 105 H z
Q104 Hz
L J
100
10
1000
Temperature (K)
2
300K
0
a.
5
a
•2
£
03
O
•4
6
8
2
4
6
8
10
12
14
16
log (f, Hz)
Figure 6.9: (a) V ariation o f sim ulated crAWDP w ith tem perature for frequencies indicated
using Eqn. (5.7). T he values o f fitting param eters are Vm=6000 K, Vo=3000 K, Ao=5
K and
to= 2 x 10*13 sec.
(b) Predicted frequency dependence o f Oadwp at 300K.
112
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
35Li20 -3 A l2 0 3 -lP :0 5-61Si0;
Li vibration
10'°
1011
1012
1013
10u
Frequency (Hz)
Figure 6.10: Far infrared conductivity data o f lithium silicate glass at RT. The shoulder
around 5x 1 0 12 Hz is the Li* ion vibrational frequency. T he resonant peaks at ~ 1 0 lj Hz
and above are the vibrations o f other structural units in the glass system .
113
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
3 5 % L i,0 61% SiO- 3% AL0_ 1 % P ,0
RT
E
o
RT
o
3
■a
c
o
III b
RT
I—
250]
200
T50I
TRl
7K
10° 101 102 103 104 105 106 1 07 10s 10s 10 1010 11 1012101310u
Frequency (Hz)
Figure 6.11: C onductivity as a function o f frequency for lithium silicate glass. Region I
represents frequency-independent dc conductivity, Region II represents the U D R region
w ith slope s-0 .5 -0 .7 , Region Rla is the NCL region w ith s—1.0-1.15, R egion nib is the
m icrow ave region, and Region IV is the FIR conductivity w ith s~2 and resonant peaks
from vibrating ions and structural units. The dotted line is a reference w ith slope =1.0.
114
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
0.247K_O-0.753GeO,
1 0 '7
: 300K
250 K
slope=1.0
200 K
'
100
Frequency (Hz)
Figure 6.12: Frequency dependence o f ct for 0.247K.20-0.753Ge02 glass for tem peratures
from 100K. to 400K. The sym bols represent the data points. The solid line jo in in g the
points is a guide to the eye. T he dashed line is a reference slope = 1.0.
115
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1000
x=0.0023
x=0.02
x=0.2
100
a
□ A
D
D
104
105
106
107
108
109
1 0 10 1 0 11
1012
xa)/Tadc
Figure 6.13: ‘M a ste r’ plot o f norm alized conductivity, cr/adc versus xco/Tardc for three
different com positions o f KzO-GeOz glass.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
: (a) CeO ,:1% G d3"
o
A
O
—
T=336K
T=350K
T=3712K
Master curve
o
0
0
10’
10°
10- ’
10'
“ /“ onset
(b) C e 0 2:11%Gd3* crystal
102
□
C
v
O
—
T=352.5K
T=335.5K
T=373.5K
T=395K
Master curve
10°
10°
“ /“ onset
Figure 6.14: M aster plots o f (a) 1% Gd3~ doped C eO : crystal (b) 11% G dJ1’ doped CeOjThe solid line is the C M R ‘m aster’ curve according to function in Eqn. (6.4). The
sym bols are experim ental data points.
117
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Frequency (Hz)
10'
10z 103 104 10s 108 107 10® 109 10'°
•
—
o
low frequency data
Master curve
MW data
104 t
0.247K2O - 0.753GeO
T=394.7K
acj
O
C
10'3 102 10 1 10°
10'
102 103 104 10s
10« 10r
(O
/CO _
“"onset
frequency (Hz)
107
10 “
y........
' ")"«—
">
E I * low frequency data
| ------- Master curve
0.247K2O - 0.753GeO :
T=346.3K
c
c
10°
10-'
10°
10’
102 103 104 105 10s
107 10s
°^u onset
Figure 6.15: Scaling plots for 0.247K 2O - 0.753G eO 2 glass at tem peratures 394.7K and
346.3
K.. The solid line is the C M R m aster curve defined b y Eqn. (6.4). T he sym bols are
experim ental data points at low and m icrow ave frequencies.
118
R e p r o d u c e d with p e r m i s s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e rm is s io n .
0.247K200.753Ge02
340K
394.5K
£
10 •4 _
.o
(n
o
Z3
"O
c
o
O
slope=1.0
10'
Frequency (Hz)
Figure 6.16: Frequency dependence o f m icrow ave conductivity o f 0.247K 2O-0.753GeO
glass betw een 300K to 400K. The dashed line is a reference line o f slope = 1.0.
119
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
0.074K20-0.926Ge02
O
1 0 -4
o
*
O
300K
350.2K
394K
E
o
^
CO
/
qO”
Oo 2 ^ o °
$ r o0 °
o
/
13
T3
C
/
/
/
s lo p e = 1 .0
/
o
/
O
/
/
/
IQ 109
108
Frequency (Hz)
Figure 6.17: Frequency dependence o f m icrow ave conductivity o f 0.074K.2O-0.926GeO
glass betw een 300K and 400K . The dashed line is a reference line o f slope=1.0.
120
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
10-3
0 .2 4 7 ^ 0 - 0.753GeO2
10"1
(a)
E* 10'5
o
^
10-6
:!
10-7
>%
o
■o 10-3
c
o
O 10‘9
10-io
O'
O'
10-11
102
103 104 105 106 107
108
10s 1 0 10
103 104 105 106 107 10a
10s 1 0 10
F req u en cy (Hz)
0 .0 7 4 K „0 - 0 .9 2 6 G eO
>>
10
o
10-8
102
F req u en cy (Hz)
Figure 6.18: C om parison o f em pirically extrapolated low f - low T data using Eqn. (6.1)
and experim ental high f - high T data for (a) 24.7 m ol% and (b) 7.4 m ol% potassium
germ anate glasses. The open circles represent the extrapolated data at low f while solid
circles represent the experim ental data at high f. The dashed line is a least square fit w ith
slope 1.1.
121
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
0.247K20-0.753Ge02
o
o
o
o
° .....
- O .......
E
o
>7
c/3
>N
>
o
zs
■O
c
o
O
.A
A
A — A
□
□
Q""0
.a
.a—-o
.a
300
V
Q—-O-—0'
350
400
Temperature (K)
Figure
6.19:
Tem perature
dependence
of
m icrow ave
conductivity
of
0.247K 20 0 .7 5 3 G e 0 2 glass com position at 5 x l0 7 Hz, 1x10s Hz, 5 x l0 8 Hz, lx lO 9 Hz,
2 x l0 9 Hz and 3 x l0 9 Hz from bottom to top respectively. The dashed lines joining the
data points are em pirical fit to TYpow er law equation.
122
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
0.24
c
0)
c
o
Q.
X
0
0
0
0u .
o
Q.
E
0
(a) low f and low T
0.21
0.18
0.15
0.12
0.09
A
A
0.06
i
l
0
5
10
15
20
25
30
K20 mol%
1.0
(b) high f and high T
C
0.9
0
C
o
Q.
X
0
0.8
0
0.7
0l_
0
A
-1
I
7
15
20
0.6
CL
E
0
f—
0.5
0.4
J
L
0
5
10
25
30
K20 mol%
Figure 6.20: The dependence o f tem perature exponent, y on alkali concentration for
potassium germ anate glass series at: (a) low f - low T regim e [133]. The line joining the
sym bols is draw n to guide the eye. (b) high f - high T regim e. The dashed line is a
polynom ial fit to the data points.
123
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
- O - 300K
-+■■■ 340K
- o ~ 392K
_ _ - A
..--A
•A .....................A-.........
, A --------------------------------
10
/f > .\
J u
Hz
J '
f
^
....■
'. • .......
E"
-, aLJ~ .'........ -
_ q. o
=3
■a
g
f=1.015x10
’.-■a
(=5.119x10
^
c
o
O
8
Hz
" 5 ^
.-O
■.•a-
° " 4 \
‘ - O ''"
.-•3
.-O-
v B > > ~ -«-
,8
f=1 .0 37 x1 0
V "^
10 -
J
!
I
I____
9
12
15
Hz
l
i
I
18
21
24
K20(m ol% )
Figure 6.21: Com position dependence o f m icrow ave conductivity for the K^O-GeO; glass
series at 100 M Hz, 512 M Hz, and I G H z at three different tem peratures. The lines
jo in in g the data points are draw n to guide the eye.
124
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
/
1 .e+ 19 r
least square fit
/
/ •
/
CO
I
E
w/
/
/
/
c
•
o
/
/
/
/
03
/
/
g
/
1 .e+18
/
c
o
/
/
o
•
/
/
/
/
/
Q
<
/
/
/
1.e+17 -
0.1
L
i
i
1.0
I I I I I 1
__________1
-
1
10.0
Li20 (mol%)
Figure 6.22: A lkali concentration dependence o f A D W P concentration in lithium
germ anate glasses at low T and low f [133]. T he dashed line is a least square fit o f the
data with slope ~0.8.
125
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
0
5
10
15
20
25
30
K20 (mol %)
Figure 6.23: V ariation o f ADW P concentration w ith K2O content in the glass
m icrow ave frequency. Line joining the sym bols is draw n to guide the eye.
126
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CHAPTER 7 FREQUENCY AND TEMPERATURE DEPENDENCE
OF ADWP CONFIGURATIONS
In chapter 6, we saw how the A D W P m odel was able to describe the conductivity
data at m icrowave frequencies in the lithium silicate and potassium germ anate glass
system s. In this chapter, the nature o f the frequency dependence o f ionic conductivity
over a wide range o f frequency, from a few Hz to a few GHz, w ill be discussed. We shall
exam ine the im portant A D W P fitting param eters and com pare them in the low T - low f
and high T - high f regions, in order to ascertain w hether or not the A D W P structures in
the two regions are sam e. As m entioned, the im portant param eters include Vm and Am, the
cut-offs for the barrier height and asym m etry energy; N, the concentration o f ADW P
configurations; and V0, the w idth o f the distribution function. A linear tem perature
dependence o f Am, given by Am=A0*"i'T w as introduced to account for the sm all increase
in the asym m etry energy w ith tem perature. The value o f t0 is not so im portant as it only
shifts the m axim um in the conductivity to higher or low er tem peratures at a fixed
frequency and/or shifts the conductivity plateau to higher or low er frequencies at a fixed
tem perature. In addition, the dependence o f these param eters on the concentration o f
alkali atom s is discussed in this chapter. B y correlating to the changes in glass structure, a
physical understanding o f the A D W P entities that are responsible for conduction at
higher frequencies is determ ined.
A com prehensive understanding o f these experim ental observations can be
achieved first by addressing to w hat extent the ion dynam ics in glasses are determ ined by
127
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
(i) interactions betw een m obile ions and glassy netw ork, and (ii) long range interactions
betw een m obile ions. Accordingly, there are theoretical m odels focussing on the m otion
o f particles in com plex energy landscapes determ ined by the disordered netw ork
[149.150] and others focussing on the interactions am ong m obile ions [151]. Dyre et. al
came up w ith a m acroscopic m odel for ac conduction in ionically conducting disordered
solids [149], The m odel considers ac conduction in an inhom ogeneous solid as random ly
varying electrical conductivity that is therm ally activated. In other w ords, they describe
the com plete range o f frequency dependent conductivity in term s o f one single
m echanism , nam ely through therm ally activated ion hopping over a distribution o f energy
barriers that is inherently present in the glassy state. W ith approxim ations, this m odel was
able to predict both the pow er laws, i.e., w here the frequency exponent o f conductivity is
'0 .7 and goes to unity as tem perature is reduced [152]. Independent studies by Hunt
[153] are also based on sim ilar views, but at high tem peratures he differentiates the
frequency dependent conductivity to arise from m ultiple hopping. However, analysis
based on M onte Carlo sim ulations o f diffusion o f ions in ordered and structurally
disordered lattices by M aass et. al [154] and D ieterich et. al [155] show s that the presence
o f both structural disorder and coulomb interactions is needed to predict the dispersion
behavior that is w idely observed in conductivity m easurem ents. C ontrary to the above
ideas, several investigators have favored the view that the nearly constant loss region
(NCL) is due to the com plex and localized m otion o f group o f atom s [156,157,158, 101,
133], These authors have independently show n that there is evidence supporting the idea
that N C L arises due to a new kind o f atom m ovem ent.
128
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
7.1 Failure of C lassical D ielectric Loss Theories
The m icrowave dynam ic behavior o f sim ple ionic crystals can be understood in
term s o f specific defects and their interactions w ith lattice vibrations [159,160]. For
instance, the m icrow ave loss in NaCl crystal is show n to be due to com bined
contributions o f sodium vacancy hopping, dipolar relaxation o f vacancy-im purity pair
and m ultiphonon quasi-resonant processes. A ttem pts to understand the m icrowave
behavior in oxide glasses based on such m odels have not been successful due to
qualitative differences in the frequency and tem perature dependencies o f c Mw from that
o f sim ple crystals.
Early work by Stevels, Isard, and others [81-87] have show n that up to five
independent sources m ay be responsible for conduction at m icrow ave frequencies in
oxide glasses, which have been reviewed in section 2.3.5. From these classical loss
m echanism s, we conclude
that qualitatively one could consider the
microwave
conductivity, (Tm\v> o f glasses in term s o f the m igration, the vibration and/or the
deform ation
loss
m echanism s.
In
general,
the
first
m echanism
consistently
underestim ates the observed m icrow ave losses in a series o f sodium alum inosilicate
glasses [86], and therefore fails to quantitatively explain the origin o f m icrow ave loss in
oxide glasses. The vibration and deform ation loss m echanism s are inconsistent and do
not agree with the com position dependence o f m icrow ave conductivity in alum inosilicate
glasses. Therefore, in spite o f three plausible sources, the m icroscopic origin o f crMw has
rem ained unknown for oxide glasses.
129
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
There have been few studies to further explain the origin(s) o f ctmw in glasses
after the early work. Recently, theories for explaining the m echanism o f m icrow ave loss
in glasses are em erging based on the better understanding o f the m odels that have been
established for low frequency dielectric loss. For instance, Funke and co-w orkers
[161,162] have advocated that m icrowave conduction is sim ply an extension o f lo w
f r e q u e n c y s in g le io n h o p p in g (universal dynamic response region, UDR). A ny additional
contribution
is believed
to com e
from
high
frequency
far infrared
vibrational
conductivity. M icrow ave conductivity is then sim ply the short tim e observation o f the
relaxation associated w ith the hopping o f ion. To test their hypothesis i.e. to verify if ctM w
can be considered sim ply as a sum o f low frequency single ion hopping and high
frequency ion vibration conductivity, we em pirically fitted the low frequency and FIR
conductivity data using the follow ing Eqn.:
cfco) = Acon + A'com
(7.1)
w here A and A ' are em pirically fitted param eters, n=0.65 and m =2. T he first term in Eqn.
(7.1) is representative o f the ion hopping contribution (U D R ) seen at low frequencies
(H z-kH z) in m ost ionic glasses and crystals, w hile the second term is due to single ion
vibrations observed at far infrared frequencies (> 1 0 12 Hz). T he fit to Eqn. (7.1) is shown
as a dashed line in fig. 7.1 for the present lithium silicate glass. C learly the extrapolation
underestim ates the experim ental
gmw
by a factor o f -1 0 , indicating that the m echanism
o f conduction in the M W region is not an extension o f ion hopping conductivity from low
frequency or FIR vibrational conductivity from high frequency. This, therefore, suggests
that
ctmw
is not sim ply a short tim e observation o f the relaxation associated with the
130
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
hopping o f ion as believed by Funke and co-w orkers, but indicates the presence o f an
additional m echanism o f m icrow ave loss in oxide glasses. Furtherm ore, F u n k e’s model
predicts a plateau at m icrow ave frequencies and beyond [161], w hich has not been
experim entally observed in oxide glasses.
7.1.1.
Failure o f Scaling Laws
W ithin the past couple o f decades, different authors have proposed scaling laws to
describe the so called universal frequency, co (=2;tf), dependence o f ac conductivity o f
glasses and crystals, which lead to a single m aster curve irrespective o f tem perature
and/or com position. [163,164,161]. A ccordingly, the ac conductivity, spanning from a
few Hz to well into the m icrow ave region is generally w ell described by a scaling relation
o f the follow ing form [161,165,166]:
^
= FI
f
\
(7.2)
where f0 is the characteristic frequency defined in term s o f dc conductivity (cr0),
tem perature (T) and the concentration o f m obile charge carriers. Roling et al. proposed
that for a sodium borate glass xN ajO -fl-xJB iO j o f given com position, a (f) m easured at
different tem peratures could be scaled using Eqn. (7.2) w ith k = a 0T [166]. H ow ever, to
also scale w ith respect to com position, this characteristic frequency was m odified to
f0= a 0T7x, so as to incorporate the changes in the num ber density o f charge carriers. As a
result, they w ere able to collapse the ac conductivity o f sodium borate glass for
131
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
0.1<x<0.3 to a single m aster curve. In other words, there is no influence o f alkali ion
concentration on the shape o f conductivity spectra, except in the crossover region.
However, a scaling analysis perform ed on o u r potassium germ anate glass
com positions, shows distinct differences am ong the conductivity spectra, particularly
near and beyond the region o f dispersion. The scaling properties o f low frequency ionic
conductivity spectra in glasses have been studied for three potassium germ anate glass
com positions and are shown in fig. 6.13 as a ‘m aster’ plot betw een a/crj c and cax/Tcdc. In
the crossover regim e, from dc to dispersive conduction, w hich spans about two decades
o f frequency, the ‘m aster’ curve is com position dependent. This is an indication o f the
fact that as the m obile ion concentration approaches zero, the U D R region vanishes and
the dc plateau crosses over directly to the NCL region. A pparently, the scaling law for a
‘m aster’ plot is valid only if the m obile ion content is high (> 2 m ol% ), such as exam ined
by Roling et al. In such cases the ion-ion separation is sufficiently small and the
coulom bic interactions are sufficiently strong that the details o f atom ic structure and
com position do not affect the U D R behavior. On the other hand, for dilute m obile ion
concentration the glass com position and structure significantly influence the UDR
response and the scaling laws cannot consolidate data for different glasses.
Sim ilar scaling analysis on the C eO i sam ples doped w ith 1% and 11% Gd'~ also
shows differences in the conductivity spectra for alm ost tw o decades o f frequency (figure
not show n). Therefore, the failure o f form ation o f a ‘universal’ m aster curve in potassium
germ anate glasses and doped C eO : polycrystalline sam ples suggest that the low
frequency response (UDR) is indeed influenced by structural changes occuring in the
132
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
glass netw ork due to addition o f alkali ions. Subsequently, Ngai and M oynihan also
found sim ilar results in low concentration V ycor glass o f com position xN aiC H lx)[0.04B a03-0.96Si02] w ith x=0.00044 [167], w here distinct differences were observed
in the dispersion region. Sim ilar finding has also been recently confirm ed on sodium
germ anate glasses w ith 0.003<x<0.1 by Sidebottom [168], who interprets the observed
discrepancy as due to neglecting the changes in cation-cation distance with alkali
concentration.
7.1.2 A pplicability o f C M R M odel to Ionic C onductivity o f Ceram ics and Glasses
A nother scaling m odel based on the concept o f m ism atch and relaxation (CM R)
proposed by Funke for crystals and glasses [145,169], invokes the creation o f som e
m ism atch betw een the hopping ion and its surroundings (refer section 6.2.1.1). The
system can relax this original mismatch, either via a backw ard hop o f the ion, or through
the relaxation o f the neighborhood. Based on this concept, the particular shape o f the
conductivity spectra over wide frequency range could be w ell defined by a m aster curve
defined by Eqn. (6.4). Analysis perform ed using this scaling equation for 1% G dJ*-doped
C eO : crystal, 1 l% G d3'-d o p ed CeO ; polycrystal (highly defective) and x K jO O -x lG e O :
glass series (refer figs. 6.14 and 6.15) show s a large disagreem ent betw een the C M R
m aster curve and low frequency experim ental data, particularly beyond the onset o f
dispersion. In addition, the extrapolation o f the m aster curve to m icrowave frequencies
overestim ates the experim ental data by alm ost an order o f m agnitude for the potassium
germ anate glass, thereby disagreeing with the C M R m odel.
133
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Perhaps a strong evidence to the inability o f C M R theory to predict ionic
conductivity in polycrystalline and glassy m aterials arises from the fact that there is an
extensive pow er law region representing the U D R behavior w hich applies to m ore than
three decades in frequency. N ow ick et al. have extensively studied the U D R response o f
variously doped CeO i ceram ics in a w ide range o f frequency and tem perature [170], A
m aster plot o f all data for GdJ~ doped CeO : sam ple show n in fig. 7.2 supports this
observation. From the m aster plot, one can clearly see that conductivity increases with a
pow er law behavior for alm ost four decades in frequency. The existence o f such a m aster
plot shows that the universal UDR response observed in various crystals and glasses can
describe the conductivity spectra not only in a lim ited frequency range as predicted by
CM R model, but is applicable over a w ider range o f frequencies. Careful investigation o f
the pow er law regions (U D R and NCL) in various glasses and crystals by independent
groups have show n that the two ‘universal’ behaviors are quite different from each other.
In particular, N ow ick et al. have studied the conductivity behavior for doped CeCh
ceram ic by com paring the tem perature and frequency dependence o f ionic conductivity in
these two regions. T heir studies gave conclusive evidence that the tw o processes, viz.
UDR and NCL are distinct from each other, so that in an interm ediate range o f frequency
and tem perature, the dispersive behavior can be analyzed only as a superposition o f U D R
and NCL [171].
Finally, the discrepancy betw een the C M R m odel and experim ental conductivity
data exists for the low defect concentration crystalline ceram ic as m uch as for high
concentration com plex glass. This m eans an im portant param eter that is dependent on ion
134
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
concentration and/or atom ic structure o f the glass has to be accounted for by the model.
Very recently, Funke has suggested that a second term should b e added in Eqn. (6.4) to
account for the structural properties o f the environm ent, but its actual form needs to be
established before analysis can be conducted further.
The failure o f the scaling m odel indicates that the C M R function described in
Eqn. (6.4) cannot predict ionic conductivity in defective crystalline and glassy materials.
In short, the contention that ion dynam ics in glasses can be described by a single hopping
m echanism over a w ide frequency range fails. The results o f the scaling analysis show
that the CM R function cannot predict conductivity at m icrow ave frequencies. This
im plies a MW is not sim ply a short tim e observation o f the low frequency UDR response,
but indicates the presence o f another m echanism o f m icrow ave loss in oxide glasses.
7.2 T he ‘Jellyfish ’ M echanism at M icrow ave frequencies
Interestingly, the m icrow ave conductivity o f oxide glasses at RT and above
behaves sim ilar to the low frequency conductivity at low tem peratures; in that both show
a linear frequency dependence, cr~co:’ , w here s '=1.0, and a w eak tem perature dependence.
From the sim ilarity o f conductivity at low T (4K -150K ) - low f (Flz-kHz) and high T
(300K-400K.) - high f (~GFIz), it is tem pting to believe that both observations have a
com m on origin. H sieh and Jain w ere the first to report this sim ilarity in lithium silicate
glass [132], w hich w as then followed by sim ilar observations b y B elin et al. on A g-doped
Ag:S glasses [80].
135
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
O ne can clearly see from figs. 6.7a and 6.18 that in both alkali silicates and
germ anates, the tem perature and frequency dependence o f a at low tem perature-low
frequency and high tem perature-high frequency regions can be reasonably well described
by a single empirical relation. In other w ords, w e are able to predict the frequency as well
as the tem perature dependence o f m icrow ave conductivity solely using the low -frequency
low -tem perature conductivity data. Therefore, the m echanism o f conduction should be o f
the sam e origin in these two regions. In other w ords, the high frequency, high
tem perature dielectric loss in oxide glasses can be established to originate from jelly fish ­
like localized fluctuations o f atom s in asym m etric double well potential configurations.
C om puter sim ulations o f ionic conductivity using the com plete ADVVP form alism
given by Eqn. (5.7) was done to establish the origin o f conduction m echanism at
m icrow ave frequencies. As described in sections 6.1.2.1 and 6.2.2.1, the optim um values
o f the fitting param eters obtained from sim ulations at m icrow ave frequencies, nam ely
V m, A0, and V0 are 6000K , 5K, and 3000K respectively for both oxide glass system s
(refer Table 6.2). The calculated value o f t 0 is 2 x l 0 '13 sec for the binary silicate glass and
5 x l0 ‘" sec for the binary germ anate glass system . Such low values for t 0 com pared to the
values obtained at low T - low f, have also been observed for jellyfish structures in
phosphosilicate and flurozirconate glasses at low tem peratures using nuclear spin
relaxation m easurem ents, w here instead o f a localized single ion hopping process, an
excitation o f ADW P configurations form ed by several atom s is observed [104,136].
Since the value o f t 0 obtained from sim ulations in the tw o glass system s are higher than
that observed for a single ion vibration (~-10'u sec), it supports the idea that the
136
R e p r o d u c e d with p e r m i s s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e rm is s io n .
conduction at M W frequencies is caused by fluctuations o f a group o f atom s rather than
isolated ions. The data on aliovalent doped (ScJ~, G d'~ and Y J_) polycrystalline CeCh
[102,103.172] have show n that there is a gradual transition from dipolar-like loss
behavior to A D W P-type behavior as the doping concentration increases. The observation
o f the anom alous low tem perature loss peak in ScJ+ doped ceria was show n by m olecular
dynam ics sim ulation to be due to the movem ent o f a cluster o f S cJ* ions and oxvgen-ion
vacancies, resulting in a distorted oxygen cage surrounding the isolated scandium ion
[I ” 3]. Lu and Jain have also reported a sim ilar observation o f low tem perature peak in
conductivity in some low concentration binary germ anate glasses [133], and attributed
the peak to arise due to jellvfish-like fluctuations o f group o f atom s. This evidence
supports the fact that the jellyfish structures are indeed com prised o f clusters or group o f
atoms.
Com parison o f the experim ental data with the m odel is show n in the form o f both
frequency and tem perature dependence o f conductivity for 24.7 m ol% potassium
germ anate glass in figs. 7.3a and 7.3b. Reasonable agreem ent betw een the data and
model provides a strong support for our hypothesis that the conduction m echanism o f
alkali oxide glasses in the m icrow ave region and elevated tem peratures originates from
l o c a liz e d f lu c tu a tio n s o f j e l l y f i s h s tr u c tu r e s .
To determ ine i f the sam e low-T ADW P configurations are responsible for high T
- high f conductivity, let us com pare the ADW P fitting param eters used in the present
analysis with those at low T and low f o f Lu [130]. The values o f the fitting param eters at
low T are V m=1000K, Ao=5K, Vo=730K and t o= l x l 0 ^ sec, w hich are different from
137
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
those obtained for high T - high f region. Since the optim um A D W P param eters,
particularly V m, V0 and t 0 in the two regions differ significantly, the ADW P
configurations that are responsible for RT - high f conductivity m ust be different from
those observed at low T-low f. This is not surprising, since different kinds o f ADW PCs
have been observed at different tem perature and frequency regim es. For exam ple, Kanert
et al. identified three kinds o f A D W PC s in lithium silicophosphate glass [174], Since
different jelly fish structures are possible, we conclude that the RT - high f ADW P
configurations responsible for a ^ w differ from those at low T and low f.
Further evidence that supports the idea that A D W PCs in the two regions are
different is seen in the tem perature dependencies o f ionic conductivity o f these two
regions. At low T and low f, the tem perature dependence follows a w eak pow er law, a(T )
x T \ where the exponent y typically lies betw een 0.08 - 0.25 (see Table 6.1), whereas at
high T and high f, y typically lies betw een 0.5 - 0.9 (see Table 6.1) for the lowest to
highest concentration o f potassium content in our glasses. O ne possible reason for this
discrepancy in the tem perature exponent can be due to the contribution o f additional
m echanism such as low frequency single ion hopping (U D R region) on CTmw- T o establish
this possibility, the m agnitude o f U D R contribution to conduction at m icrowave
frequencies w as calculated using a ‘m aster’ plot o f norm alized conductivity, a / a dc against
cd/T cjjc
and then subtracted from the observed
cmw -
The U D R m echanism follows an
ArThenius tem perature dependence w ith activation energy Eac=(l-s)-Edc, w here s is the
frequency exponent with values typically betw een 0.5-0.7 in m ost oxide glasses and
crystals, and EdC is the d.c activation energy [175]. As a result o f this subtraction, the
138
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
tem perature exponent y decreases from 0.86 to 0.16 for the 24.7 m ol% K iO glass, but
remains unaffected for the rest o f the glass series as seen in fig. 7.4. This indicates that
the single ion hopping influences the tem perature dependence o f ctmw only for 24.7 mol%
K: 0 glass for w hich dc conductivity at room tem perature is the highest w ithin the present
glass series. For com positions x < 20 m ol% , dc conductivity at R T is low enough that the
L DR ion m ovem ent does not influence the m icrow ave conductivity.
W ithin the confines o f our high T - high f m easurem ent range, the dependence of
the tem perature exponent 'y ' on alkali ion concentration is show n in fig. 6.20b. Since the
tem perature exponent is related to the asym m etry energy, A, betw een two w ells o f the
ADW P structure, the variation o f y (hence A) w ith alkali ion concentration indicates the
dependence o f A D W P configurations on structural changes in the glass netw ork as alkali
ion concentration is changed. From fig. 6.20b, we see that y rapidly decreases as
potassium ion concentration changes from 0.23 m ol% to 2 m ol% . It appears that this
rapid decrease is due to the form ation o f com pact 3-m em bered [GeOa] rings instead o f 6m em bered rings [176,177]. T he reduction in the ring size provides slightly m ore volum e
for the atom s to m ove. If w e w ere to consider only the localized m ovem ent, this slight
volume increase w ould cause a decrease in asym m etry o f A D W P structures. A fter 2
m ol% , y increases rapidly first up to 5 m ol% and then very slow ly up to 20 m ol% , after
w hich there is again a rapid increase. The sudden increase in y a fter 2 m ol% is likely due
to the form ation o f [G e 0 6] units, w hich are structurally less com pact than the [GeCU]
units. Finally, for concentration >20 m ol% , the form ation o f N B O s in the glass netw ork
results in the increase o f asym m etry in A D W P structures.
139
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
7.3 C om position dependence o f M icrow ave C onductivity
In order to ascribe som e physical description and/or characteristics to the jellyfish
structures, it is useful to examine the com position dependence o f ctmw and A D W P fitting
param eters. From the com position dependence o f ctmw (refer fig. 6.21), we can see that
conductivity increases w eakly w ith potassium concentration. In general, the greater the
num ber o f alkali ions present, higher is the m icrow ave loss, and hence conductivity. The
com position dependence o f conductivity at m icrow ave frequencies follows a weaker
pow er law dependence w ith alkali ion concentration than that observed at low T and low
f by Lu et. al. For exam ple, the conductivity (above 2 m ol% ) at low T and low f varies
proportional as ctLt l f * -t0 ' 9 [133], whereas at m icrow ave frequencies,
ctm w
x -t°4. .x
being the K.:0 concentration as shown in fig. 7.5. This w eak com position dependence o f
cmw indicates that the effective perim eter o f jellyfish structures (size o f jellyfish)
becom es larger w ith alkali oxide addition, since m ore alkali ions are required at
m icrow ave frequencies to cause the same change in ctmwThe variation o f the Cmw
t
with alkali concentration show s no correlation to the
dependence o f dc conductivity on glass com position, w hich is established to be due to
single ion hopping. In other words, the dependence o f Cmw
t
on alkali content in the glass
is not related to dc activation energy changes but rather to variations in the different
ADWP
quantities.
From the sim ulation results o f A D W P
equation, w hich
sum m arized in Table 6.2, we can see that the param eters V m, V0 and
with alkali ion concentration. This confirm s the fact that
ctmw
t0
are
do not change
is not related to changes in
140
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
activation energies, since Vm refers to the potential barrier o f the A D W P structure.
Therefore, the com position dependence o f ctmw should arise due to variation o f N or Am
with alkali content in glass. On com parison, we find that the com position dependence o f
N follows a sim ilar pow er law dependence as
ct M w
(refer figs. 6.21 and 6.23). Hence the
changes in the total num ber o f A D W P configurations in the netw ork due to the addition
o f alkali ions plays an im portant role in determ ining the com position dependence o f ctmwU nlike in germ anate glasses, in silicates, ct<ic and ctmw follow a m onotonic change
with com position. For exam ple, in sodium alum inosilicate glass series for fixed N a and
with varying AI2O 3, T opping and Isard found the dielectric loss to m onotonically increase
with Al increase until A l/N a ratio was equal to unity [178], the sam e as found for the dc
conductivity o f this glass system . The non-m onotonic behavior is com m on am ong
germ anates leading to the anom aly seen in m any physical properties as a function o f
alkali concentration.
In the lithium germ anate series, Lu determ ined at low tem peratures that the
.ADWP concentration increases w ith .r m uch m ore rapidly than
that in sodium
alum inosilicate and sodium borosilicate glasses, where the conductivity (and ADW P
concentration) increases w ith the concentration o f bridging oxygens (BO) [130]. It is
known that in alkali germ ante glasses, the initial addition o f up to 15-20 m oI% alkali
oxide m ostly converts GeO.» to G e 0 6 units and produces NBOs to a sm all extent, there is
no report o f increase in BO concentration [36]. We note from data analysis that the
.ADWP concentration in K -G e glasses at m icrow ave frequencies is about a few 1020 cm"'
(see Table 6.3), but the alkali ion concentration is about 1022 - 1023 cm"3. This m eans,
141
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
only a fraction o f alkali ions contribute to jellyfish-type conduction in the M W region,
presum ably those that are co-ordinated to the netw ork atom s in som e particular
configuration as illustrated in figure 7.6. T he sublinear dependence o f ctmw on
com position (oc .r0'4) suggests that m o r e th a n o n e alkali ion effectively contribute to one
jellyfish structure, w hich also includes the netw ork form er atom s. Thus, alkali ions have
an im portant role in ionic conduction at m icrow ave frequencies.
C om paring the values o f N obtained from data fitting to the A D W P m odel at M W
region in the two glass system s, we find that the values are consistently higher than the
values at low f - low T. For instance, the value o f N is alm ost an order o f m agnitude
higher in the silicate glass and alm ost two orders o f m agnitude higher in the germ anate
glass com positions than the corresponding low T - low f values. In other w ords, there are
m ore jellyfish structures per unit volume at high f - high T than at low f - low T. From
the calculated values o f N, the spacing betw een two neighboring A D W P structures at
m icrow ave frequencies com es to approxim ately - 1 0 0 A in these oxide glasses. This
distance is the effective m axim um possible region o f jellyfish influence at m icrow ave
frequencies. H ow ever, considering that w e treat jelly fish to be non-interacting w ith each
other, the size o f the atom cluster should be sm aller w hen com pared to the spacing
betw een two neighboring AD W P structures, typically - 2 5 A o r less.
S u m m a ry
The differences observed in AD W P fitting param eters betw een low f - low T and
high f - high T conductivity suggest that the m echanism s in these two regions, even
142
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
though o f a com m on origin, m ay have som e differences in the constitution o f the jellyfish
species responsible for conduction. That is, the constitution o f A D W P species that are
responsible for conduction at high frequency - high tem perature is different from their
low frequency - low tem perature counterparts. Furtherm ore, the variation in m icrowave
conductivity (hence jellyfish structures) w ith glass com position is m ainly due to changes
in interm ediate range structure involving netw ork atom s and alkali ions. At m icrowave
frequencies, the effective size o f the jellyfish is roughly 25 A or less.
7.4 Connection to Far Infrared C onductivity
It is im portant to realize up to w hat frequencies the RT conductivity is dom inated
by the 'jellyfish' m echanism , because ultim ately at far infrared frequencies localized
vibrations o f individual cations are observed. The connection betw een the m icrowave and
far infrared conductivity is analyzed in term s o f the relative contribution o f
ct, \ d w p
to the
observed conductivity before the onset o f localized vibrations from individual cations.
We can see from fig. 7.7 that the m agnitude o f Gadwp (show n as dashed-dotted line) at
frequencies greater than - 1 0 11 Hz is m ore than an order o f m agnitude less than o fir and,
hence, will not contribute to conductivity at such frequencies. H ow ever, the contribution
o f ctfir to cr in the frequency range betw een 109 to 1011 Hz is significant and the
com bined contribution o f
ctadw p
and
c t f ir
is show n as open circles in fig. 7.7. The total
conductivity increases sm oothly from the ADW P region o f slope 1.0 to a region o f slope
2.0, where we observe the vibrational loss. Since the localized vibrational frequency o f
the Li" ion is betw een ~ 7 x l0 12 Hz (230 c m '1) and - I x l O 13 Hz (380 c m '1) in such high
143
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
alkali silicate glasses [179], and between ~ 2 x lO i: Hz (75 c m '1) and ~ 4 .5 x l0 12 Hz (140
c m '1) for the K.' ion in the germ anate glass series [180], the low -frequency flank o f the
far infrared spectrum w ould consist o f a transition from localized m otion o f ADW P
structures to the local vibration o f single alkali ions.
A nother point o f interest is the high frequency plateau observed in som e fast ion
conducting crystals and glasses. This change in the frequency dependence o f conductivity
near - 1 0 10 Hz has been interpreted in term s o f a change in which an ion ju m p s without
coupling to its environm ent [181]. Theoretical prediction o f high frequency plateau in Na
(3-alumina has been extensively exam ined by Ngai in term s o f the coupling m odel [181]
and Funke by ju m p relaxation model [182]. Based on these m odels, the plateau from
uncoupled hops in our silicate glass should be less than the actual conductivity by two or
m ore orders o f m agnitude. The ADW P model also predicts a high frequency plateau, but
its onset frequency and m agnitude are m uch higher than those predicted by the coupling
m odel for this specific glass system .
144
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
102
35LioO 61 SiOo 3AloOq 1PoOi
101
10 °
?
.o
10-1
w
10-2
•
o
v
RT
low frequency
far infrared
microwave
2.0
• I ' 10-3
S
10-4
c
1 0 ’5
O
1Q-6
o
396.2K,
10-7
10*
10'9
1 0 1 102 103 104 105 1 0 6 107 108 109 1 0 1° 1 0 111 0 1210 131 0 14
Frequency (Hz)
Figure 7.1: Real part o f conductivity as a function o f frequency o f a lithium silicate glass.
The sym bols are experim ental data a t : • - low [133]; V - m icrow ave; and O - far infrared
frequencies. The dashed line is the em pirical fit to Eqn. (7.1) using low frequency and
FIR data. T he dashed-dotted lines are em pirical extrapolation o f Cmw
t
to high frequencies.
145
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Figure 7.2: N orm alized conductivity versus cot for G d3* doped C eO i at three doping
levels [170].
146
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
0.247K2O-0.753GeO>
;
•
j
■ T=390K
f
T=300K
-j0-4 L
/
ADWP mode!
e
,0
* *
>*
o
3
■o
c
o
O
A
■
10-5
108
109
Frequency (Hz)
•
■
f=i00 MHz
f=i GHz
0.247K 20 -0 .7 5 3 G e 0 2
ADWP model
|
104
o
3
■o
c
o
O
10-5
300
350
400
Temperature (K)
Figure 7.3: Com parison betw een experim ental m icrow ave conductivity (sym bols) and
AD W P m odel (dashed line) for 0.247K 20 -0 .7 5 3 G e 0 2 glass.
147
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
24.7%K20 - 75.3%Ge02
20%K2O - 80%GeO2
L
r
E 1 .e-4 tCO
® 9
^
9
$
..................................
/
*
§ 1 e-5 L•O
c
o
O
1.e-6 t-
1 .e-6
280
1.e-7
280
320
360
400
320
360
1
400
T e m p e ra tu re (K)
T e m p e ra tu re (K)
7.4%K20 - 92.6%Ge02
0.23%K2O - 99.77% G e02
1.e-4
,o
CB 1 e-4 -
s
i
© o
o o o o o o o
_j
o o
1 .e - 6 r ^
i
*=
1.e1
o o o o
*
r
§ 1.e-7 ko
p
1.e-8
l
r
1.e-8
280
J________I
320
360
1.e-9
280
400
T e m p e ra tu re (K)
Figure 7.4: Influence o f calculated
oudr
320
360
3
400
T e m p e ra tu re (K)
on experim ental
g Mw
at different tem peratures
for four different com positions o f K.-Ge glass. The open circles are the experimental
m icrowave conductivity, the solid circles are m agnitudes o f calculated U D R conductivity
at m icrowave frequencies, and diam onds are conductivity values after subtraction o f ctudr
from C mw-
148
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
0-9
3e-4
f
O '10 -
E
.o
CO
LL
h
0 - 11 L
2e-4
o - 12 -
O’13
r
O*14
0-15
E
cj
a
a
t?
c/)
A
0-16
0 -1 7
H 1e-4
------------------------- 9e-5
—o— a dc at RT
- 8e-5
—-a —at RT |' 7e-5
q-13
0 -1 9
0-20
•o-
0-21
0-22
o'
3
crLTLFat10K H 6e-5
</
5e-5
5
10
15
20
25
30
K20 (mol%)
Figure 7.5: C om position dependence o f dc conductivity at RT, low T - low f conductivity
at 10K [133] and m icrow ave conductivity at RT. The sym bols are experim ental data
points. Lines joining the data points are draw n to em phasize the trend.
149
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
M
M
0
O
O
Ge
.
O
\
/
Gc ^
\
/
Ge >
o
O
0
o
o
/
O
° \ Ge/
\
>Ge
\ 0 '
M
Ge
•
M
Ge
/
Ge
\
0
M
Ge
0
O
O
O'
0
O
o
V
0
O*
M*
O
\
/
.W o ;
\
M
\Ge
Ge
Figure 7.6: Suggested structure o f jelly fish configuration at m icrow ave frequencies,
com prising o f netw ork atom s (G e and O ) and alkali ions (M~). T he shaded region is the
area o f jelly fish influence, typically ~25 A or less.
150
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
E
—
•
ADWP model
FIR extrapolated
ADWP+FIR
MW
2-0y
c/5
0
11
12
13
Frequency (Hz)
Figure 7.7: C ontribution o f ADW P conductivity to high frequency conductivity in
3 5 L i;0 -3 A l2 0 3 -lP ;0 5 -6 1 S i0 2 glass at RT. T he dashed-dotted line is the sim ulated
.ADWP conductivity (Eqn. (5.7)) w ith fitting param eters as indicated in the text. The solid
sym bols represent experim ental data points at M W (circles) and FER (triangles)
frequencies. The open circles represent the com bined contribution o f
ctadwp
and
ctfir
the 109 - 10u Hz range. The solid line joining open circles is draw n to guide the eye.
151
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
in
CHAPTER 8 CONCLUSIONS
A com prehensive m odel has been presented to describe the conductivity o f alkali
oxide glasses at m icrow ave frequencies. M icrow ave conductivity o f lithium silicate glass
and potassium germ anate glass series, as m easured by the transm ission line m ethod using
a netw ork analyzer in the frequency range from 50 M Hz to 3 GH z and tem peratures from
room tem perature (RT) to 400K, shows nearly linear frequency dependence with weak
therm al activation. Em pirically we have dem onstrated that the low T (5K <T<150K ) low f ( lOHz-lOOkHz) and high T (>RT) - high f(50M H z-few GHz) conductivity in these
oxide glass system s are related to each other. Therefore, w hen the tem perature is low or
when the time scale is too short to observe single ion hopping, conductivity arises from
very sim ilar 'jellyfish-like' m otion o f a group o f atom s over an asym m etric double well
potential (.ADWP). Com puter sim ulations using the ADW P form alism describe the linear
frequency dependence o f microwave conductivity well. In addition, the m odel also
predicts successfully the tem perature dependence o f o\iw.
C om parison o f the AD W P param eters nam ely, m axim um barrier energy Vm,
width o f distribution function V0, relaxation tim e x, and A D W P concentration N, at low T
- low f and high T
- high f show significant differences in their values in the two
regions. Furtherm ore, the tem perature dependence o f conductivity at M W region is
stronger than that observed at low T and low f, w hich is m anifested as higher values o f
tem perature exponent ‘y \ The differences in the param eters o f high T - high f and low T low f conductivity im ply that even though the m echanism o f conduction in the two
152
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
regions has a com m on origin, there exist differences in the constitution o f the ADWP
entities responsible for conduction.
From the variation o f A D W P param eters w ith alkali concentration, we see that
Vm, V0 and t 0 do not change significantly w ith the addition o f alkali atoms. Thus these
param eters are not related to the details o f structure o f the glass netw ork. On the other
hand, the concentration o f A D W PC s varies with alkali concentration in a m anner similar
to m icrow ave conductivity, thereby indicating
that changes
in ctmw with glass
com position is m ainly due to changes in concentration o f A D W P configurations in the
glassy netw ork. From data analysis, it is established that only a fraction o f alkali ions
contribute to jellvfish-type conduction in the MW region, particularly those that are co­
ordinated to the netw ork atom s. The effective region o f influence o f each jellyfish
com prising o f netw ork atom s and alkali ions is roughly - 2 5 A or less. The sublinear
dependence o f
ct M w
on com position
( ctm w
*
.t0 4 )
suggest that m ore than one alkali ion
effectively contribute to one jelly fish structure, w hich also includes the netw ork former
atoms.
Finally, by com paring the sim ulation results w ith the experim ental data in the
MW and FIR regions, it is found th at the conductivity in the frequency region o f 109- I 0 n
Hz can be explained in term s o f com bined contributions from m ulti-atom jellyfish
excitations and single ion vibrations.
153
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
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[130] X. Lu. Ph.D. D issertation Thesis, Lehigh U niversity, 1994.
[130] W .K. Lee, J.F. Liu and A.S. N ow ick, P h y s . R e v . L e tt., 67, 1991, pp: 1559.
[131] A.S. Now ick, A.V. V aysleyb, H. Jain and X. Lu, E lectrically Based
M icrostructural Characterization, eds. R.A . G erhardt, S.R. T aylor and E.J. Garboczi,
Mat. Res. Soc. Sym p. Proc. 411, 1996, pp: 99.
161
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
[132] C.H. Hsieh, H. Jain, J. N o n - C r y s t. S o lid s , 203, 1996, pp: 293.
[133] X. Lu, H. Jain, J. P h y s . C h em . S o lid s , 55, 1994, pp: 1433.
[134] K.S. Gilroy, W .A. Philips, P h il. M a g . B , 1981, pp: 735.
[135] H. Sieranski, O. Kanert, M. Backens, U. Strom , K.L. N gai, P h y s. R e v . B , 47, 1993,
pp: 681.
[136] S. Estalji. 0 . Kanert, J. Steinert, H. Jain, K.L. Ngai, P h y s . R e v . B 43, 1991, pp:
7481.
[137] j . a . K apoutsis, E.I. Kam itsos, G.D. Chryssikos, Y.D. Y iannopoulos. A.P. Patsis.
C h im ic a C h r o n ic a . N e w S e r ie s , 23, 1994, pp: 341.
[13S] X. Lu, W.C. H uang and H. Jain, P h y s. C h em . G la s s e s , 37, 1996, pp: 201.
[139] D.L. Price. A.J.G. Ellison, M.L. Saboungi, R.Z. Hu, T. Egam i. W.S. Howells. P h ys.
R ev. B . 55. 1997. pp: 11 249.
[140] E . I . Kam itsos, Y .Y iannopoulos, H. Jain and W .C. Huang, P h y s . R e v . B . 54, 1996.
p p : 9775.
[141] H. Verweij, J.H .J.M . Buster, J. N o n -C r y s t. S o lid s , 34, 1979, pp: S i.
[142] W.C. Huang, H. Jain, G. M eitzner, / . N o n - C r y s t. S o lid s , 196, 1996, pp: 155.
[143] X. Lu, Ph.D. D issertation, Lehigh University, 1994.
[144] H. Kanht, B e r. B u n s e n g e s . P h y s .C h e m ., 95, 1991, pp: 1021.
[145] K. Funke, B. Roling and M. Lange, S o lid St. I o n ic s , 105, 1998, pp: 195.
[146] B. Roling, S o l i d S t. I o n ic s , 105, 1998, pp: 185.
[147] B. Roling, A. Happe, K. Funke and M.D. Ingram . P h y s . R e v . L e tt., 78, 1997, pp:
2160.
162
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
[148] K. Funke and D. W ilmer, M a t. R e s. S o c . S y m p . P r o c ., 548, 1999, pp: 403.
[149] T. Riedel, J.C. Dyre, J. N o n - C r y s t. S o lid s 172-174,1994, pp:
1419.
[150] A. Hunt, J. N o n -C r y s t. S o lid s 160, 1993, pp: 183.
[151] K. Funke, B. Roling, M. Lange, S o l i d St. I o n ic s 105, 1998, pp: 195
[152] J.C. Dyre, P h y s. R e v. B 48, 1993, pp: 12511.
[153] A. Hunt, J. N o n - C r y s t. S o lid s 183, 1995, pp: 109.
[154] P. M aass, M. M eyer and A. Bunde, P h y s . R e v . B 51, 1995, pp: 8164.
[155] W. D ietrich, P. Pendzig, S o l i d S t. I o n ic s 105, 1998, pp: 209.
[156] A.S. Now ick, B.S. Lim and A.V. V aysleyb, J . N o n - C r y s t. S o lid s 172-174, 1994,
p p : 1243.
[157] U. Strom , K.L. Ngai, O. Kanert, J. N o n - C r v s i. S o lid s 131-133, 1991. pp: 1011.
[158] K.L. N gai, 0 . Kanert, H. Jain, J. N o n - C r y s t. S o lid s 222, 1997, pp: 383.
[159] B. Vleng, B.D. Klein, J.H. Booske, R.F. Cooper, P h y s. R e v. B 53, 1996, pp: M i l l .
[160] M. Sparks, D.F. K ing and D.L. M ills, P h y s . R e v. B 26, 1982, pp: 6987.
[161] K. Funke, C. Cram er, B. Roling, T. Saatkam p, D. W ilmer, M.D. Ingram , S o lid
S ta te I o n ic s 85, 1996, pp: 293.
[162] C. C ram er, K. Funke, B e r. B u s e n g e s . P h y s . C h e m . 96, 1992, pp: 1725.
[163] D. L. Sidebottom , P.F. Green and R.K . Brow, J. N o n - C r y s t. S o lid s 203, 1996, pp:
300.
[164] B. R oling, A. Happe, K. Funke and M .D. Ingram, P h y s. R e v . L ett. 78, 1997. pp:
2160.
[165] H. K ahnt, B e r. B u n se n . G es. P h y s . C h e m . 95, 1991, pp: 1021.
163
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
[166] B. Roling, M .D. Ingram , M . Lange and K. Funke, P h y s . R e v . B 56, 1997, pp:
13619.
[167] K.L. Ngai and C.T. M oynihan, M a te r . R es. S o c . B u ll., 23, 1998, pp: 51.
[168] D. L. Sidebottom , P h y s . R e v . L e tt. 82, 1999, pp: 3653.
[169] K. Funke and D. W ilm er, S o l i d St. I o n ic s 136-137, 2000, pp: 1329.
[170] A.S. Now ick, A.V. Vaysleyb and I. K uskovsky, P h y s . R e v . B 58, 1998, pp: 8398.
[171] A.S. Now ick, B.S. Lim and A.V. V aysleyb, J . N o n - C r y s t. S o lid s 172-174, 1994,
p p : 1243.
[172] R. Gerhardt. W .K. Lee, and A.S. Now ick, J. p h y s . C h e m . S o l i d s , 48, 1987, pp: 563.
[173] A.N. Corm ack, C.R. Catlow and A.S. Nowick, J. P h y s. C h e m . S o lid s , 50, 1989, pp:
177.
[174] O. Kanert, R. Kuchler. J. Peters, A. Volm ari, H. Jain, H. Eckert, E. Ratai, J. N o n C iy s t. S o lid s , 222, 1997, pp: 321.
[175] K.L. Ngai and R.W . Rendell, P h y s . R e v. B , 38, 1988, pp: 9987.
[176] Y.D. Y iannopoulos, E.I. Kam itsos, H. Jain, in Physics and A pplications o f N onC rvstalline Sem iconductors in O ptoelectronics. NA TO A dvanced Research W orkshop,
M oldova, 1996.
[177] H. Jain, W .C. Huang, E.I. Kam itsos, Y.D. Y iannopoulos, J . N o n - C r y s t. S o lid s ,
222, 1997, pp: 361.
[178] J.A. Topping, J.O . Isard, P h y s . C h e m . G la s s e s , 12, 1971, pp: 145.
[179] J.A . Kapoutsis, E.I. K am itsos, G.D. Chryssikos, Y.D. Y iannopoulos, A .P. Patsis,
M. Prassas, C h im ic a . C h r o n ic a , N e w S e r ie s 23, 1994, pp: 341.
164
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
[ 180] E.I. K am itsos, Y.D. Y iannopoulos, H. Jain and W .C. Huang, P h y s . R e v . B , 54,
1996, p p : 9775.
[181] K.L. Ngai, U. Strom , P h y s. R e v . B 38, 1988, pp: 10 350.
[182] K. Funke, B. Roling, M. Lange, S o l i d St. i o n i c s 105, 1998, pp: 195.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
A P P E N D IX A
C O A X I A L T R A N S M I S S IO N L I N E - S C H E M A T IC S
(a) C e n te r C o n d u cto r
0.12 in
\
0.059" dia.
2.4 in
2.6 4 in
(b) O u ter C o n d u cto r
0.27 in
f
2.4 in
166
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
A P P E N D IX B
S T E P - B Y -S T E P I N S T R U C T I O N O F M I C R O W A V E
M E A S U R E M E N T U S IN G N E T W O R K A N A L Y Z E R H P 8 7 5 3 C
I)
Initialization:
(a) Load program ‘EPSM U 3’ on the com puter and open the initialization menu
for netw ork analyzer HP8753C.
(b) Initialize “C A LIBRA TIO N T Y P E ’ to full two port calibration, ‘VELOCITY
FA C T O R ’
to 0.666,
“C H A R A C TER ISTIC
•FR EQ U EN C Y R A N G E’ o f interest,
and
IM PE D A N C E ’
‘SW EEP
to
50Q,
T Y P E ’ (typically
logarithm ic sweep).
II)
C alibration
of
N etwork A nalyzer
(H P8753C )
using calibration
kit
(H P 85031B ):
(a) Calibrate port 1 for reflection (S n ) by connecting ‘O P E N ’, ‘SH O R T’ and
‘L O A D ’ successively to port 1.
(b) Repeat step (a) for port 2 (for S?:)(c) C onnect port 1 and 2 to form a ‘T H R U ’.
(d) C alibrate for transm itted coefficients, S 12 and S21(e) O m it the ‘ISO LA TIO N ’ calibration routine and press ‘D O N E ’ key on the
calibrate m enu.
( 0 O nce the above standards have been calibrated, the V N A calculates the error
coefficients for accurate dielectric m easurem ent.
167
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
I ll)
S-Param eter and D ielectric Perm ittivity M easurem ent:
(a) Place sam ple inside the coaxial airline (sam ple holder) such that it is flush
w ith one end o f the holder. Connect the sam ple holder betw een the two port
cables o f the netw ork analyzer.
(b) U sing the program ‘E PS M U 3’, click on ‘A C Q U IR E D A T A ’ to m easure the
four S-param eters data in the form o f real and im aginary parts. Save the data
set in a file, < f ile n a m e .s p > .
(c) Go to ‘SETUP C A L C ’ m enu in the program and input M EA SU REM EN T
TY PE (transm ission/reflection, non-m agnetic, one sam ple), and accurate
values for AIRLIN E LEN G TH , A IRLIN E T Y PE (coaxial), INNER AND
O U TER DIAM ETERS OF TH E C O A X IA L A IR L IN E, SA M PLE LENGTH,
INNER AND O U TER D IAM ETERS O F TH E SA M PLE, DISTA NCE OF
THE SAM PLE FROM PO RT letc., along w ith their tolerances.
(d) Use ‘LA BO R A TO RY ’ m enu to set conditions like tem perature, hum idity,
pressure etc. Also input the INITIA L V A LU ES for the dielectric perm ittivity
o f the sam ple if known.
(e) Then click on ‘CA LC R E SU LT S’ m enu in the program to calculate the
dielectric perm ittivity (s' and s") o f the sam ple after loading the s-param eter
file < file n a m e .s p > . T he program now perform s an iterative procedure to
calculate the dielectric constant and loss o f the sam ple under test as a function
o f frequency.
(f) O nce calculation is com plete, save the files for plotting.
168
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
APPENDIX C RESEARCH PUBLICATIONS
1) S. K rishnasw am i, H. Jain, A.S. N ow ick, “A nom alous frequency dependence o f
electrical conductivity o f low alkali concentration germ anate glasses” , C e r a m ic
T r a n s a c tio n s 88, 353 (1997).
2) H. Jain, S. K rishnasw am i, “Com position dependence o f frequency pow er law o f ionic
conductivity o f G lasses”, S o l i d S ta te I o n ic s 105, 129 (1998).
3) S. Kxishnaswami, H. Jain, E.I. Kam itsos, J. K apoutsis, “C onnection betw een the
M icrow ave and Far Infrared Conductivity o f O xide G lasses” , J o u r n a l o f N o n - C r y s ta llin e
S o lid s 274, 307 (2000).
4) H. Jain, S. K rishnasw am i, O. Kanert, R. Kuehler, “N ew C onduction M echanism at
M icrow ave Frequencies o f a Silicate G lass”, C e r a m ic T r a n s a c tio n s 106, 351 (2000).
5) S. K rishnasw am i, H. Jain, O. Kanert, “Com position and T em perature dependence o f
Ionic C onductivity at M icrow ave Frequencies o f Potassium G erm anate G lasses” ,
presented at the A m erican Ceram ic Society Conference held in St. Louis ‘00.
6) S. K rishnasw am i and H. Jain, “ ’Jellyfish’ m echanism o f m icrow ave conductivity o f
com plex m aterials”, P oster presentation at the M eta-M aterials W orkshop organized by
The D efense A dvanced Research Project Agency (D A R PA ) in Sept. 2000.
7) S. K rishnasw am i, H. Jain and O. Kanert, “Jellyfish M echanism o f C onduction at
M icrow ave Frequencies in Alkali O xide Glasses”, presented and subm itted to the
A m erican C eram ic S ociety Conference held in April 2001.
8) S. K rishnasw am i, A.S. N ow ick, H. Jain, “A pplicability o f the C M R M odel to AC
Ionic C onductivity o f C eram ics and G lasses”, to be subm itted to S olid State Ionics.
169
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
VITA
Sum ithra Krishnaswam i was bom to Suiochana and Srinivasan K rishnasw am i on
D ecem ber 27, 1971 in M adras, India. U pon com pleting her high school in 1989, she
passed a national exam and gained entry into PSG C ollege o f Technology, Coim batore,
India. She obtained her B achelor’s degree in A pplied Sciences in 1992 and M aster’s
degree in M aterials Science in 1994. In pursuit o f higher know ledge in the field o f solid
state, she passed a national level exam to gain adm ission into M aster o f Technology
program in Solid State T echnology at Indian Institute o f Technology, M adras, India.
There she spent 18 m onths 'understanding’ solid state physics before being accepted into
the Ph.D program in M aterials Science & Engineering at Lehigh University, Bethlehem.
PA in 1996. Since then she has been at Lehigh, w orking on her Ph.D dissertation am ong
other things.
170
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
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