close

Вход

Забыли?

вход по аккаунту

?

Breakdown of superconductivity in high transition temperature oxide superconductors: Microwave studies

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI
films the text directly from the original or copy submitted. Thus, some
thesis and dissertation copies are in typewriter face, while others may
be from any type of computer printer.
The quality of this reproduction is dependent upon the quality of the
copy submitted. Broken or indistinct print, colored or poor quality
illustrations and photographs, print bleedthrough, substandard margins,
and improper alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete
manuscript and there are missing pages, these will be noted. Also, if
unauthorized copyright material had to be removed, a note will indicate
the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand comer and
continuing from left to right in equal sections with small overlaps. Each
original is also photographed in one exposure and is included in
reduced form at the back of the book.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher quality 6” x 9" black and white
photographic prints are available for any photographs or illustrations
appearing in this copy for an additional charge. Contact UMI directly
to order.
A Bell & Howell information Company
300 Norm ZeeO Road. Ann Arbor. Ml 48106-1346 USA
313/761-4700 800/521-0600
ABSTRACT
T itle o f D issertation:
B reak d ow n o f S up ercon d u ctivity in H igh T ransition
Tem perature O xid e Superconductors — M icro w a v e S tudies.
J yotsn a Sridhar R am achandran, D o cto r o f P h ilo so p h y , 19 9 5 .
D issertation directed by:
Satindar M . B hagat
P rofessor o f P h y sic s
D epartm ent o f P h ysics
T w o fu ndam ental m easureab le properties o f the ideal su p ercon d u ctin g state
are its ab ility to e x p e l flu x from w ithin its bulk and its infinite c o n d u c tiv ity . In the
n ew h igh transition tem perature su p erco n d u ctin g (H T S C ) o x id e s th ese p rop erties
can be a ffe c te d sig n ific a n tly by granularity. For e x a m p le , granular ( m icr o n -siz ed
p o w d ers and a g g lo m er a tes) H T S C m aterials are w e ll k n ow n to ex h ib it ab sorp tion
o f m ic r o w a v e p o w er w h en su b jected to a dc m a g n etic field o f o n ly a fraction o f a
m T.
T h is p o w e r ab sorp tion c o m e s as a su rp rise c o n s id e r in g that fo r an id eal
su p ercon d u ctor, o n e e x p e c ts the m a g n etic field w ith in the bulk o f the m aterial to
vanish w h ich m ean s tbat from the point o f the both the m ic r o w a v e s as w ell as the
dc m a g n e tic fie ld no p o w er ab sorp tion sh o u ld o ccu r.
S ev er a l o f th e p ro p o sed
m o d els attem pt to accou n t for this lo ss by assu m in g the p resen ce o f flu x w ith in the
bulk o f the sa m p le.
H o w e v e r , the in v e stig a tio n s o f static m a g n etic re sp o n se o f
th ese p o w d ers, carried out in this laboratory, clea rly ind icated that, contrary to this
a ssu m p tio n , the bulk o f the sa m p le d o es support zero m a g n etic in d u ctio n .
T he
f o c u s o f th is stu d y , th e r e fo r e , h as b een to u n d ersta n d th is b re a k d o w n in
s u p e r c o n d u c tiv ity that b e c o m e s apparent in the p resen ce o f m ic r o w a v e s and to
re co n cile this w ith the ob servation o f B = 0. T h is report is, to our k n o w le d g e , the
o n ly on e o f its kind w here a d etailed study o f the field ind u ced m icro w a v e resp on se
in p o w d e r H T S C sa m p les has been undertaken o v e r a w id e range o f m ic r o w a v e
fr eq u e n c ie s, tem perature and m agn etic field . W e sh o w that th e o b serv a tio n s can be
m o d e lle d by c o n sid e r in g the w e a k lin k s p resen t in th e gran u lar s p e c im e n as
r e s is tiv e ly sh u n ted J o sep h so n ju n c tio n s (R S J s).
T h e R SJ m o d el is sh o w n to
a c c o u n t for th e fr e q u e n c y as w e ll as th e tem p eratu re d e p e n d e n c e s o f the
p h en om en on . T he e ffe c t o f c y c lin g the sam ple through the dc m agn etic field results
in h y steretic m icr o w a v e absorption. T h is report a lso co v e rs a d eta iled study o f the
fr e q u e n c y , tem perature and d c field e ffe c ts on the h y ster etic a b sorp tion .
O n ce
again, w e sh o w that a sim p le ex ten sio n o f the RSJ m od el used to accou n t the initial
absorption can a lso acccou n t for the hysteretic effects.
Breakdown o f Superconductivity in High Transition Temperature
O xide Superconductors — M icrowave Studies.
by
Jyotsna Sridhar Ram achandran
D issertation subm itted to the Faculty o f the Graduate S ch ool
o f T he U n iversity o f M aryland in partial fu lfillm en t
o f the requirem ents for the d egree o f
D o cto r o f P h ilosop h y
1995
A d visory C om m ittee:
P rofessor Satindar B hagat, C h airm an /A d visor
P rofessor E llen W illiam s
P ro fesso r A rnold G lick
P rofessor A lic e M ign erey
A ssista n t P rofessor Fred W ellsto o d
UMI Number: 9607812
UMI Microform 9607812
Copyright 1995, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
300 North Zeeb Road
Ann Arbor, MI 48103
D E D IC A T IO N
T o the m em ory o f m y parents
S hanta and D , V. Rajan
ACKNOW LEDGEM ENTS
First o f all I w o u ld like to ex p ress m y sin cere gratitude to m y a d visor. Prof. S.
M . B h a g at w h o se con stan t w ords o f en co u ra g em en t saw m e through th is project. H is
b rillian t in sig h ts into the p rob lem (w h ich m ore o ften than not ca m e to h im in the w ee
hours o f the m orn in g! ) m ad e all the d iffer en ce. I w o u ld a lso lik e to thank Prof. S om
T y a g i w h o m ade in valu ab le con trib u tion s to th is w ork. H is u n fla g g in g en th u sia sm and
h is th o u gh tfu l c r iticism h elp ed in n u m erou s w a y s. T h e c o n v e r sa tio n s (a c a d e m ic and
o th e r w ise ) I had w ith Dr. V . S esh u B ai w a s sp e c ia lly en rich in g and a lw a y s h elp ed m e
see things in better light. T hanks are a lso due to Dr. M arc M an h eim er o f the Laboratory
o f P h ysical S c ie n c e s for both the u n lim ited a cc ess o f his laboratory and the m any liv ely
d is c u s s io n s I h a v e en jo y e d w ith him .
N e e d le s s to sa y I h ave learnt m o st from m y
fe llo w stu d en ts S a m L o fla n d and M in g H u an g w h o p a tien tly tried to a n sw er every
(absurd?) q u estion I m an aged to c o m e up w ith. L ife through graduate sch o o l co u ld not
h a v e b een as p le a sa n t if it w a s n ’t fo r m y fr ie n d s - S a y o k o and K en F ord , Jutta
L u e ttm er-S tr a tm a n n , A n n a M a th a i, D r. M . R a je sw a r i, C h u h e e K w o n , S an tan u
B hattach arya, E d d ie B e ll, Sri D a s, A n n a and K evin A n d rew s, D heeraj S a n g h i, Sharat
C handran, Pratim a and A run S im h a , to nam e a fe w . S p e cia l thanks are due to M rs.
B h a g a t w h o s e w arm c o m p a n y (n ot to m e n tio n th e fo o d ) m y fa m ily and I a lw a y s
en jo y ed .
M o st im p ortan tly I w o u ld lik e to thank m y fa m ily in In dia for the u n relen tin g
support through the years. B ut b efore I c lo s e , I have to accep t that I w ou ld n 't b e sitting
here w ith a c o m p le te d th esis and k e y in g in the a c k n o w le d g e m e n ts if it w asn't for my
h u sb an d Sridhar R am achandran w h o is at the m o m en t tryin g to ou t d o m y d aughter,
M alavik a, in ch eerin g m y efforts!
TABLE OF CO NTENTS
S ection
P a ge
L ist o f T a b le s ..................................................................................................................................................vi
L ist o f F ig u r e s ................................................................................................................................................ vii
C hapter I.
1 .1
I n tr o d u c tio n
to
S u p e r c o n d u c t iv it y ................................................................1
State o f Perfect C on d uctivity and P e r f e c t ............................................................
D ia m a g n etism ....................................................................................................................7
1.2
T herm od yn am ics o f the S uperconducting S t a t e .................................................8
1.3
L ondon E q u a tio n s.............................................................................................................11
1 .4
P e n e t r a t io n
1.5
BCS
1 .6
C on d u ctivity in the S up ercon du ctin g State — T w o - F lu id
D e p t h ..................................................................................................... 12
T h e o r y .................................................................................................................. 13
M o d e l.................................................................................................................................... 15
1.7
L andau-G inzburg T h e o r y ............................................................................................. 17
1.8
T y p e-I and T yp e-II S u p er co n d u cto rs......................................................................18
1 .9
P in nin g in T ype-II S u p er co n d u cto r......................................................................... 22
1 .1 0
A n isotrop y in H T SC O x id e s ....................................................................................... 2 6
1.11
Josep h son J u n c tio n ........................................................................................................ 2 8
1 .1 2
R e sistiv e ly S hu n ted J osep h son Junction ( R S J ) ................................................ 3 6
1 .1 3
Surface Im p e d a n c e .......................................................................................................... 38
Chapter II
M e th o d ..............................................................................................................................4 0
2 .1 .
S a m p l e s ..............................................................................................................................4 0
2 .2
M icrow ave S p ectro m eter..............................................................................................4 4
2 .3
C r y o g e n ic s ..........................................................................................................................4 8
2 .4
Experim ental T e c h n iq u e ............................................................................................... 4 9
C hapter III
R e s u l t s .............................................................................................................................5 4
3.1
D C M a g n e tiz a tio n a n d S u s c e p t ib ilit y ........................................... ..............55
3 .2
Z ero-F ield M icrow ave A b so r p tio n .......................................................................... 64
3 .3
F ield Induced M icrow ave A bsorption - Virgin
C u r v e ....................................................................................................................................81
3 .4
U n iversal Virgin C urve - D is c u s s io n ...................................................................... 105
Chapter IV
H y s te r e s is ......................................................................................................................... 132
iv
S ection
4 .1
Page
R e s u lts ...................................................................................................................................................133
4 .2 D i s c u s s i o n ................................................................................................................................. 149
Chapter V
S u m m a ry ...........................................................................................................................170
R e f e r e n c e s .................................................................................................................................................... 1 7 2
v
LIST OF TABLES
N u m b er
I . 1.
Page
E stim ated penetration depths A, and coh eren ce len gth s for so m e
H T S C o x id e s .....................................................................................................................26
II. 1.
L ist o f sa m p les and their respective grain d iam eters........................................ 43
III. 1.
L ist o f sam p les used in the m agnetization and su scep tib ility
s t u d y ..................................................................................................................................62
111.2 .
Z ero-K elvin L ondon penetration depth o f p ow d er H T S C
s a m p le s
fr o m
p r e v io u s
s t u d ie s .......................................................................6 7
111.3.
L ist o f sam p les used in the m icrow ave s tu d y ......................................................8 0
111.4 .
Sam p le characteristics and parameters o f V C ......................................................9 4
IV . 1.
T hreshold field valu es as a function o f sam p le tem perature at
10 G H z ..................................................................................................................................147
vi
LIST OF FIGURES
N u m b er
P age
1.1
M a g n etiza tio n
2
1.2
F ield in d u ced m icr o w a v e ab sorp tion at 10 G H z and 77 K .
6
1.3
Path in d ep en d en t nature o f flu x e x p u lsio n in the su p erco n d u ctin g
9
state.
1.4
T he cross se c tio n o f an iso la ted vertex in type-11 su p ercon d u tor
20
1.5
B ean m od el for a slab.
25
1.6
S ch em a tic diagram o f a unit c e ll o f Y B a 2C u 30 7 (Y B C O )
27
1.7
Id ea lized form o f th e i-v cu rve for an S IN ju n c tio n
29
1.8
C u rren t-voltage ch aracteristics for a Jo sep h so n ju n ctio n
29
1.9a
S ch em a tic o f a current d riven SIS tunnel ju n c tio n
30
1 9b
C r o ss se c tio n o f a J o sep h so n ju n ctio n .
33
1 .1 0
Fraunhofer d iffraction pattern o f th e J o sep h so n critical current.
35
1.11
Current d riven r e sistiv e ly sh un ted J o sep h so n ju n c tio n circuit.
37
2.1
H istogram o f p a rticle s iz e con stru cted for a Y B C O p ow d er.
42
2 .2
S ch em a tic o f the m icr o w a v e sp ectrom eter
46
2 .3
S ch em a tic o f the exp erim en tal setup to gen erate and m easu re lo w dc
47
m agn etic field .
2 .4
S ch em a tic o f apparatus to c o o l the sam p le.
50
2 .5
Z ero field m icr o w a v e transition in a layered thin film
52
3.1
Z FC isoth erm o f the 10 p m Y B C O p ow d er at 4 .2 K.
57
3 .2
Z FC m a g n etic isoth erm in 5 p m B S C C O p o w d er at 7 7 K.
58
3 .3
Z FC isoth erm o f th e 5 0 p m Y B C O a g g lo m era te sam p le at 4 .2 K
59
3 .4
Initial su sc e p tib ility o f the 10 p m Y B C O p ow der.
63
3.5
Z ero field m icr o w a v e transition s o f m icron siz e d Y B C O grain s at 10
68
G H z.
vii
N u m b er
3 .6 a
P age
Ferrel fit to ze ro -fie ld m icr o w a v e transition o f the 2 p m (grain d ia .)
70
Y B C O p o w d er at 10 G H z.
3 .6 b
Ferrel fit to ze r o -fie ld m icr o w a v e transition o f th e 6 p m (grain d ia .)
71
Y B C O p o w d er at 10 G H z.
3 .6 c
Ferrel fit to z e ro -fie ld m icr o w a v e transition o f the 10 p m (grain d ia .)
72
Y B C O p o w d er at 10 G H z.
3 .7 a
Z ero -field m icr o w a v e transition in P rB C O /Y B C O /P rB C O sa n d w ich
75
film fit to th e L on d on m o d el at 10 G H z.
3 .7 b
Z ero -field m icr o w a v e transition o f A K Z O (2 p m ) at 10 G H z fit to
76
the L on d on m o d el.
3 .8
Z e r o -fie ld m icr o w a v e tran sition o f 5 0 p m Y B C O a g g lo m era te at 10
77
G H z fit to th e L on d on m o d el.
3 .9
Z ero field m icr o w a v e transition o f 5 p m B S C C O p o w d er at 10
G H z.
78
3 .1 0
Z ero fie ld m icr o w a v e tran sition o f 10 p m Y B C O p o w d er at 10
G H z.
79
3.11
S ch em a tic o f V C ab sorp tion.
83
3 .1 2
V C , 10 p m Y B C O p o w d er at 10 G H z, 5 0 K .
84
3 .1 3
V C , 10 p m Y B C O p o w d er at 10 G H z, 4 .2 K .
85
3 .1 4
V C , 5 p m B S C C O p o w d er at 2 2 G H z, 7 7 K.
86
3 .1 5
V C , 5 0 p m Y B C O p o w d er at 2 2 G H z, 7 7 K.
87
3 .1 6
V C , 10 p m Y B C O p o w d er at 2 .5 G H z, 7 7 K.
88
3 .1 7
V C , 2 p m Y B C O p o w d er at 2 2 G H z, 7 7 K.
89
3 .1 8
V C , Y B C O p ellet at 10 G H z, 2 8 K.
90
3 .1 9
V C , 5 0 p m Y B C O p o w d er at 3 6 G H z, 77 K.
91
3 .2 0
V C , 10 p m Y B C O p o w d er at 3 6 G H z, 77 K.
92
3.21
U n iv ersa l V irgin C urve.
98
3 .2 2
T em perature d ep en d e n c e o f H0 for m icron siz e d p o w d er sa m p les.
99
3 .2 3
T em perature variation o f
a
for 10 p m Y B C O p o w d er sa m p le at 10
G H z.
viii
100
N u m b er
P age
3 .2 4
T em perature d ep en d en ce o f a (T ) /a ( 0 ) for Y B C O p ell et at 10 G H z.
101
3 .2 5 a
F req u en cy d ep en d en ce o f a for 10 p m Y B C O sa m p le at 77 K.
102
3 .2 5 b
F req u en cy d ep en d en ce o f a for 5 0 p m Y B C O a g g lo m era te sam p le
103
at 7 7 K.
3 .2 5 c
F req u en cy d ep en d en ce o f a for 5 p m B S S C O sa m p le at 7 7 K.
104
3 .2 6
W eak link m o d elled as an RSJ.
113
3 .2 7
C a lcu la ted V C .
117
3 .2 8
D e p e n d en ce o f field param eter x 0 o n q 0.
120
3 .2 9 a
T em perature d ep en d en ce o f param eter H 0 for m icr o n -siz ed p ow d ers.
122
3 .2 9 b
T em perature d ep en d en ce o f param eter H 0 in th e ca se o f p ellets.
123
3 .3 0
F req u en cy d ep en d en ce o f H 0
126
3 .3 0 a
T em perature d ep en d en ce o f a (T )/a (0 ) for 10 p m Y B C O p o w d er at
130
10 G H z fit to the RSJ m o d el.
3 .3 0 b
T em perature d ep en d ece o f a (T ) /a ( 0 ) fo r 10 p m Y B C O p o w d er at 10
131
G H z fit to th e R SJ m od el.
4.1
S ch em a tic o f the h y steresis lo o p o f lo w -fie ld in d u ced m ic r o w a v e
134
ab sorp tion in a granular H T SC sam ple.
4 .2
E ffect o f dc field sw e e p rate and m axim al field PoHmax for 10 p m
135
Y B C O p o w d er at 10 G H z and 4 .2 K.
4 .3 a
H ysteretic L F IM A in the region R+ for the 10 p m Y B C O p ow d er at
138
10 G H z, 4 .2 K.
4 .3 b
H y steretic L F IM A in the region R q for 10 p m Y B C O p o w d er at 10
139
G H z, 7 7 K.
4 .3 c
H ysteretic L F IM A in the region R. for 10 p m Y B C O p o w d er at 10
140
G H z, 4 .2 K
4 .4
F ield span d ep en d en ce o f 6 H in the region o f R+.
141
4 .5
E ffect o f the m axim al field on the h y steresis loop .
142
ix
N u m b er
4 .6
Page
Irreversib le R esid u al A b sorp tion v ersu s PoH max fo r th e
10 p m
143
S H as a fu n ction o f p 0H max for the 10 p m Y B C O p o w d er at 3 6 G H z
144
Y B C O p o w d er at 10 G H z and 7 7 K.
4 7
and 4 .2 K.
4 .8
5 h a s a fu n ctio n o f | i 0H max for th e 10 p m Y B C O p o w d er at 10 G H z
145
and 4 .2 K.
4 .9
4 .1 0
T em perature variation o f th e th resh old field , p 0H Th at 10 G H z.
T h e local field versu s the ap p lied dc field in the re g io n R+ for th e
148
10
153
p m Y B C O p o w d er at 10 G H z, 7 7 K.
4.11
A sy m m etr y o f th e R+ h y ster esis lo o p .
154
4 .1 2
L ocal field v a lu e as a fu n ction o f PoH max for the 10 p m Y B C O
155
p o w d er at 10 G H z and 7 7 K.
4 .1 3
L ocal field v a lu e as a fu n ction o f p 0H max f ° r th e 10 p m Y B C O
156
p o w d er at 10 G H z and 50 K .
4 .1 4
C a lcu la ted lo o p s for the reg io n Ro-
159
4 .1 5 a
L ocal field ca lcu la ted u sin g the RSJ eq u a tio n s for th e 10 p m Y B C O
160
p o w d er at 10 G H z and 7 7 K.
4 .1 5 b
L o ca l field ca lcu la ted u sin g the R SJ eq u a tio n s for th e 10 p m Y B C O
161
p o w d er at 10 G H z and 5 0 K.
4 .1 6
4 .1 7 a
C a lcu la ted h y ster esis lo o p for th e reg io n R..
163
S c a lin g o f th e lo ca l field w ith the field param eter p 0H 0 at 10 and 3 6
164
G H z for th e 10 p m Y B C O p o w d er at p 0H max = 2 2 m T and 7 7 K.
4 .1 7 b
S c a lin g o f th e lo ca l field w ith th e field param eter PoH0 at 10 and 3 6
165
G H z for the 10 p m Y B C O p o w d er at PoHmax = 21 m T and 5 0 K.
4 .1 8
S ch e m a tic o f th e h ysteretic critical current d en sity lo o p .
X
169
Chapter I.
Introduction
Prior to 1986, the p h en o m en o n o f su p e rco n d u c tiv ity w a s c o n fin e d
to lo w tem p eratu res, i.e ., b e lo w 2 5 K. W ith the d isc o v e r y [1] o f su p e rco n d u c tiv ity
in B a -L a -C u -O b y B ed n o rz and M u ller at 35 K , interest in th ese rare earth o x id e s
soared and su p ercon d u ctivity in m any other o x id e sy stem s w as d isc o v ere d .
A ll the su p ercon d u ctors k n ow n so far fall into o n e o f the fo llo w in g
tw o
m ain c a te g o r ie s
- (1 ) typ e I w h ich are so ft su p erco n d u cto rs or (2 ) typ e II,
w h ich can be eith er a so ft or hard su p ercon d uctor. T h e d istin ctio n o f hard or soft
su percon d uctor refers to its ab ility to pin m agn etic flux and w ill be d isc u sse d further
in the s e c tio n s b elo w .
A n im portant d ifferen ce b etw een typ e-I and type-II su percon d u ctors
is as fo llo w s . T h e su percon d u ctin g state is ch aracterized by tw o unique properties 1) zero re sista n c e and 2) zero m a g n etic in d u ction , i.e ., B = 0 . In oth er w o r d s, an
id eal su p e rco n d u c to r is a p erfect co n d u c to r as w e ll as a p erfect d ia m a g n e t. T he
s e c o n d p rop erty is k n o w n as the M e is s n e r -O c h s e n fe ld e f fe c t, n a m ed after its
d isc o v ere rs. A typ e-I su percon d u ctor is ch aracterized by a sin g le critical fie ld , H e,
a b o v e w h ich m a g n etic flu x enters the bulk o f the m aterial and the su p erco n d u ctin g
state is d estr o y ed . T h e m a g n etic isoth erm in a typ ical typ e-I sa m p le is sh o w n in
F ig. 1. la . T h e c a s e o f the hard or typ e-II su p ercon d u ctors is m ore c o m p le x . Flux
b e g in s to en ter the m aterial at the lo w er critical field H c l and th e m a g n etiz a tio n
d ecrea ses w ith further increase in field . A t H c 2 , the upper critical
iai
Type
- I
Superconductor
NORMAL
STATE
5
M EISSN ER
STATE
I b l type - I I
'
MIXED
Su per conductor
ST AT E
Bc2
APPLIED
MAGNETIC
INDUCTION,
B/Ttsla
F ig, 1 .1 M a g n etiza tio n as a fu n ction o f the extern ally ap p lied d c m a g m etic field for
(a) a ty p e-I su p erco n d u cto r.
B c rep resen ts the critical field for flu x entry, (b ) a
ty p e-II su p erco n d u cto r, w h ere B C 1 and B c 2 are the lo w er and upper critical field
re sp e ctiv ely .
fie ld , s u p e r c o n d u c tiv ity is c o m p le te ly d estr o y ed .
ty p ica l
T h e m a g n e tic iso th erm o f a
ty p e-II su p erco n d u cto r is sh o w n in F ig. 1.1b.
B e tw e e n H c i and H C2 the
sa m p le e x is ts in a "mixed" state, i.e ., it con tain s region s that are su p ercon d u ctin g as
w ell as parts that are norm al.
T h e n e w H T S C m a teria ls (r a re-ea rth -b a sed and b ism u th -b a se d
c o p p e r o x id e s ) c o m e u n d er th e ty p e -II c a te g o r y .
T h e ir a b ility to su sta in
s u p e rco n d u c tiv ity under h igh m a g n etic field s (H c 2 > 100 T ) m a k es th ese m aterials
p o ten tia lly te c h n o lo g ic a lly very im portant. H o w ev e r, a th orou gh u nderstanding o f
their p rop erties is not q u ite at hand. O n e o f the p ro b lem s that m a k es the picture
co m p lica te d is the inherent granular nature o f th ese m aterials. G ranularity is know n
to d o m in ate their electrical and m agn etic respon se. It is o f particular con cern in the
ca se o f the m icr o w a v e resp on se w h ich has p roven to be sen sitiv e to the m icro sco p ic
structure o f the sam p le.
T h e f o c u s o f th is w ork is to e x a m in e m ic r o w a v e a b so r p tio n ,
o b ta in e d from su rfa ce im p ed a n c e m ea su rem en ts, in p o w d e r and sin tered H T S C
sa m p le s.
It is w e ll k n o w n [2, 3 ] that in a su p e rco n d u c tin g m aterial, the o n set o f
su p erco n d u ctiv ity is sig n a lled by a rapid d ecrease in the m icr o w a v e p o w er absorbed
in ze ro a p p lie d field ; in th e su p e r c o n d u c tin g state th e a b so rp tio n is n e g lig ib le .
H o w e v e r , as has b een d em o n stra ted by variou s grou p s [4 - 8 ], in granular H T S C
s a m p le s , at T < T c , an in c r e a se in th e m ic r o w a v e p o w e r a b so rp tio n b y th e
su p erco n d u cto r can be brought about by ap p ly in g an extern al dc m a g n etic field o f
o n ly a fra ctio n o f a m T .
T h is in c rea se in the lo s s in d ic a te s a b rea k d o w n o f
su p erco n d u ctiv ity . T he o b servation is intrigu in g b eca u se static fie ld m easu rem en ts
for the sam e field region s indicate the sam p les to be w ell w ithin the su percon d ucting
3
r e g im e .
It h as to be r e c o g n iz e d , h o w e v e r , that static fie ld m e a su r e m e n ts, in
g e n e r a l, y ie ld a r e sp o n se w h ic h is sp a tia lly a v era g ed o v e r th e sa m p le w h ile
m icr o w a v es are sen sitiv e to the d etails o f the sam ple structure. It then turns out that
the u se o f m ic r o w a v e s as a probe g iv e s on e the ab ility to in v estig a te the breakdow n
o f su p ercon d u ctivity in th ese granular H T S C sp ecim en s.
T h e q u estio n to be an sw ered by th is th esis is, w hat is the ca u se for
the b rea k d ow n o f su p e rco n d u c tiv ity in th ese granular H T S C sa m p les?
A lth ou gh
sev era l m o d els [9 - 11] h ave b een p rop osed to accou n t for this p h e n o m en o n , as w e
shall sh o w , no sin g le o n e is able to accoun t for the variety in the o b serv a tio n s. T he
lack o f a g o o d u n d erstan d in g o f the data has largely b een due to the fact that, up
until n o w , variation s due to param eters su ch as tem perature, m ic r o w a v e freq u en cy
and the ap p lied field w ere not all m ap p ed out. W e h ave n o w carried ou t a d etailed
study to inclu d e all th ese effe c ts w h ich has en ab led the d ev elo p m en t o f a reasonably
su c c e ssfu l m od el to exp lain the o b served features o f the p h en om en on .
O u r in v e s tig a tio n in c lu d e s a s y s te m a tic stu d y o f m ic r o w a v e
ab sorp tion in sa m p les m ad e o f ran d om ly orien ted grain s o f varied s iz e s as w e ll as
so m e o rien ted sa m p le s both in ze ro and lo w m a g n etic fie ld s
(H d c < 2 5 m T ). It
turns ou t that the p h e n o m en o n o f the dc fie ld -in d u ce d m ic r o w a v e ab sorp tion has
v a s tly d iffe r e n t c h a r a c te r istic s in p o w d e r s and oth er gran u lar
s p e c im e n s as
co m p a red to thin film s . W h ereas the lo ss in thin film s is linear up to 1.5 T , n onh y ste r e tic , and im m e a su r a b ly sm a ll fo r T < 7 7 K, in th e c a s e o f the granular
sa m p les a sig n ifica n t lo s s is seen in field s < lm T , the lo ss is clea rly n on -lin ear (se e
F ig . 1. 2) and p e r sistis d o w n to 4 .2 K. In ad dition the p h e n o m en o n e x h ib its rich
te m p e r a tu r e , m a g n e tic fie ld and m ic r o w a v e fr e q u e n c y d e p e n d e n c e s .
4
The
in v e s tig a tio n s w e r e ca rried ou t o v e r a broad ran ge (2 G H z to 6 0 G H z ) o f
m ic r o w a v e fr e q u e n c ie s
and te m p era tu re s in th e ra n g e 4 .2 K to T c . T h e
p h e n o m e n o n , it turns ou t, can be c a te g o r iz e d under i) the virgin a b sorp tion , i.e .,
a b soprtion on first ex p o su re to ap p lied field (V irgin C u rve - V C ) and ii) h ysteretic
ab sorp tion that fo llo w s the V C w h en the sam p le is su b jected to a c y c lic m a g n etic
field .
T h e fie ld -in d u c e d lo s s in the film s is g en era lly agreed to b e due to
v o rtic es and is fairly w e ll u n d erstood [1 2 ]. H o w ev e r, as w e shall d em on strate, o n e
ca n n o t u se vortex o sc illa tio n to acco u n t for th e lo ss in the p o w d ers. R ather, it is
w e ll k n o w n [1 3 - 15] that w e a k lin k s d o m in a te th e r e sp o n se o f gran u lar
s u p e r c o n d u c to r s .
As
we
s h a ll
sh o w ,
th e
cau se
fo r th e
b reakdow n
of
su p e rco n d u c tiv ity is the result o f fie ld -in d u c e d p o w e r ab sorp tion by w e a k lin k s
present in the sam p le. T he resistiv ely shunted Josep h son ju n ction (R S J) m od el first
p ro p o sed by A . D u lc ic et al. [1 6 ] for granular su p ercon d u ctors has b een d e v e lo p e d
[1 7 , 18] to ex p la in the fie ld -in d u c e d virgin ab sorp tion in clu d in g its freq u en cy and
tem p era tu re d e p e n d e n c e s .
A ls o , an e x t e n s iv e in v e s tig a tio n [1 9 , 2 0 ] o f the
h y steresis p h en om en on w a s carried out and the p rop osed w eak link m o d el has been
e x ten d e d to y ie ld the h y ster esis lo o p s as w e ll. T o our k n o w le d g e , th is is the o n ly
report o f its kind in th e literature d e ta ilin g th e v a rio u s fie ld and tem p eratu re
d ep en d e n c ie s o f the h y steresis loop s.
5
0.
thin film
pwdr
e
3
%
c
_o
w
&
0.1
o
its
x>
<
i-H
u
£
o
Cl,
0.0
0.0
0.1
0.2
0.3
0.4
0.5
DC field (T)
F ig . 1.2.
F ie ld in d u c e d m ic r o w a v e p o w e r ab sorp tin at 10 G H z and 7 7 K in
Y B C O p o w d er and thin film sa m p le s.
P N represen ts the lo s s in the norm al state.
T h e arrow s ind icate the d irection o f the dc field sw eep .
6
T h is introduction w ill first r e v ie w o f s o m e o f the properties o f the
co n v en tio n a l su p ercon d u ctors and then d isc u ss fe w a sp ects o f the typ e-II m aterial
relev a n t to the p resen t stu d y . D e ta ils ab out the sa m p le s and the ex p e r im e n t is
d is c u sse d in C hapter II. T h e results o f the virgin absorption and the d eta ils o f the
R S J m o d e l are in c lu d e d in C h a p ter III.
O b se r v a tio n s o f h y s te r e s is in th e
p h en o m en on is d isc u sse d in C hapter IV. C o n clu sio n s are presented in C hapter V.
1.1
State of Perfect Conductivity and Perfect
D iam agnetism
S u p e r c o n d u c tiv ity is said to o cc u r w h en the
sa m p le on c o o lin g
b e lo w a cr itica l tem p eratu re (T c ) a c h ie v e s a state o f zero r e sistiv ity , i.e. p = 0 .
M e issn e r and O ch se n fe ld [21] d isc o v e r e d that in this state o f p erfect c o n d u c tiv ity ,
the sa m p le on ap p lica tio n o f a m a g n etic field d e v e lo p s a su rface current so as to
o p p o se the penetration o f any ap p lied m agn etic flu x. In this ca se B = 0 in the bulk o f
the sa m p le. A su p ercon d u ctor th erefore a lso b eh a v es as a p erfect d ia m a g n et and
th is is gen erally regarded as the m ore fundam ental property o f a superconductor.
For a d ia m a g n et (for c o n v e n ie n c e , w e c o n sid e r an in fin ite sa m p le),
since
B = p n(H + M ) = 0
( 1 .1 )
w h ere p 0M is the m agn etization , p 0H - the ap p lied fie ld and p 0 = 4 7 r x 10 7
the
perm eability o f free space. T herefore M = - H and the su scep tib ility is
( 1 .2 )
7
M e issn er and O c h se n fe ld a lso sh o w e d that the d ia m a g n e tic state o f
the su p ercon d u ctor did not d ep en d on the th erm od yn am ic or m agn etic path taken to
reach the su p ercon d u ctin g state. In other w ord s, unlike a m aterial ch aracterized by
in fin ite c o n d u c tiv ity a lo n e , the d ia m a g n e tic b e h a v io r o f th e m a teria l w a s
in d ep en d en t o f
w h eth er the fie ld w a s first a p p lied at T > Tt and the sp e c im e n
c o o le d b e lo w the critical tem perature or v ic e versa (s e e Fig. 1. 3). T h is in d ica tes
that the su p ercon d u ctin g transition in m agn etic field is reversib le and h en ce ju stifies
the u se o f th erm od yn am ics to d escrib e the su percon d u ctin g state.
1.2
Therm odynam ics of the Superconducting State
U s in g the d ifferen tia l M a x w e ll relation for the G ib b 's free en erg y
per unit v o lu m e,
dG = -S d T - B • d H
w h ere S is th e en tro p y and G is th e G ib b s free e n e r g y .
(1 .3 )
F or an in fin ite ly lo n g
c y lin d e r (d e m a g n e tiz a tio n fa c to r = 0 ) w ith the a p p lied fie ld H = H z , at c o n sta n t
tem perature, w e find from E q. (1 .3 ):
H
G ( T ,H ) - G ( T ,0 ) = -J B (H ')-dH "
0
8
(1 .4 )
a)
T>Tc
, Hext = 0
T<I
, H
/O
T <T
.H„yO
’
0
b)
T>Tc
,H
exl '
F ig . 1.3. Path in d ep en d en t nature o f flu x ex p u lsio n in the su p ercon d u ctin g state. In
(a) the m agn etic field is ap plied after reaching the su percon d u ctin g state w h ile in (b)
the sphere is c o o le d to reach the su percon d u ctin g state in a m agn etic Field. T he final
state is identical.
9
In the normal state B = jU()H for material that is non-m agnetic . and therefore
G N( T , H ) - G N( T , 0 ) = - / / 0 ^ - .
(1.5)
In the su percon d ucting state B = 0 and therefore,
G S( T , H ) - G s ( T ,0 ) = 0 .
A t the critical field
(1.6)
the n orm al and the su p e rco n d u c tin g p h a se s sh o u ld b e in
eq u ilib rium , therefore,
G s ( T , H c ) = G N( T , H t )
(1.7)
w h ich y ie ld s for the con d en sation en ergy,
uJ
AG = G N( T , 0 ) - G s ( T ,0 ) = H o ~ f '
10
( 1 *8 )
1.3
London Equations
T h e d isc o v e r y o f the M e issn e r -O c h se n fe ld e ffe c t p ro v id ed a strong
c lu e to the fact that the current in a su p ercon d u ctor is lin k ed to the m a g n etic field
rather than the elec tric field . F. and H. L on d on [2 2 ] therefore id en tified the n eed to
m o d ify O hm 's law , w h ich in ordinary co n d u cto rs relates the current to the elec tric
field . T h e y d e v e lo p e d a tw o -flu id m o d el in w h ich n, the total d en sity o f electro n s,
has tw o c o m p o n e n ts - n n - the d en sity o f elec tro n s in the norm al flu id and n s - the
d en sity o f the electro n s in the su perflu id . T he superfluid electro n s are not scattered
by im purities or p hon on s and wi l l be freely accelerated by any electric field E . T he
equation o f m otion is just
m - ^ - = eE
dt
(1.9)
w here m is the m ass o f a su perelectron , v s is its v elo city and e is its charge.
T h e su p ercon d u ctin g current d en sity is by d efin ition ,
(
1. 10)
H e n ce, E q. ( 1. 9) can be w ritten as
d ( m
\
(1.11)
U s in g F araday's law
curl E
o n e obtains
=
- f i 0—
(
1. 12)
— ( cr uwrri/AJ .s 4dt
H
= 0.
(1.13)
m
F. and H. L o n d o n r e c o g n iz e d that O hm 's law as a p p lied to norm al m etals w o u ld
c?H
on ly lead to }1{) — - = 0 and not the M eissn er-O ch sen feld ob servation B = 0 . Instead
at
they intergrated Eq. (1 .1 3 ) and ch o se the particular solution
curUs + ^ ~
m
H ~ 0
( 1 .1 4 )
Eq. (1 .1 4 ) is the L on d on eq u ation .
1.4
Penetration Depth
T o o b ta in the M e is s n e r -O c h s e n fe ld c o n d itio n w e h a v e to u se
A m p ere's law
and Eq. (1 .1 4 )
J = — V xB
(1 .1 5 )
V 2B = ^ ^ - B
m
( 1 .1 6 )
to g et
E q. (1 .1 6 ) is o f the form
V iB = - y B
X
w h e r e
A 2=
( 1 .1 7 )
( 1 . 1 8 )
-------------------- r -
Man *e
12
Eq. ( 1 .1 7 ) a llo w s a so lu tio n su ch that the fie ld in sid e the su p e rco n d u c to r d ie s
e x p o n e n tia lly w ith in a d ista n ce A from the su rface s o that in sid e a sa m p le (o f
d im e n s io n s m u ch larger than A )
B = 0.
A is term ed the L on d on p en etration
d epth. T h e H T S C o x id e s , w h ich are o f c o n ce rn h ere, are h ig h ly a n iso tro p ic and
results in d ifferen t valu es for the penetration depths a lo n g the three a x es (s e e T able
1.1).
O fte n q u o te d [4 0 ] v a lu e s at ze ro tem p eratu re are Au7, = 1 6 0 0 A , w h ile
M S A ,* .
1.5
BCS Theory
T h e L o n d o n m o d e l w h ic h g iv e s a s im p le d e s c r ip tio n o f the
e lec tro d y n a m ics o f the su p ercon d u ctor is a m a c ro sco p ic th eory. In 1957 B ard een ,
C o o p er and S c h rieffer form u lated a m ic r o sc o p ic th eory, o th er w ise k n ow n as the
B C S th eory. T h e B C S theory [2 3 ] fo r su p erco n d u ctiv ity re lie s on a net attraction
b etw een e lec tro n s. T he idea origin ated from L eon C o o p er w h o first d em on strated
that if o n e a ssu m ed an attractive interaction b etw e en any tw o elec tro n s in a norm al
m etal, it resulted in dram atic ch a n g es in the properties o f the m etal. H e sh o w e d that
an attractive interaction, h o w ev e r sm all, lead s to the form ation o f electron pairs and
lead s to an u n stab le Ferm i sea. C o n seq u e n tly , pairing o f electro n s co n tin u e s, w ith
the fo rm ation o f ea ch pair red u cin g the en er g y o f the s y s te m , until it r e a ch es an
eq u lib riu m p oin t such that the b in d in g en erg y o f yet another pair eq u a ls zero. S u ch
an e lec tro n pair b ou n d in m o m en tu m sp a ce is c a lle d a C o o p er pair. T h e d ista n ce
o v er w h ich the elec tro n s in the C o o p er pair rem ain correlated is k n ow n as the B C S
co h er en ce len gth £ .
£ and th e p en etration depth A , form th e tw o fu n d am en tal
length s c a le s o f a su p ercon d u ctor. T h e en erg y required to break the C o o p er pair is
term ed the gap en er g y 2 A .
A n im p ortan t B C S eq u a tio n relates the ze ro K elv in
value o f the gap en ergy to the critical tem perature as
13
^
^
=
3.52
(
1. 1 9 )
k BT,
w h ere k„ is the B o ltzm a n n con stan t. Bq. (1 .1 9 ) y ie ld s 2A = 3 0 m eV for the pair
breaking en ergy for the H T S C m aterial ( T c ~ 100 K).
In m o st c o n v e n tio n a l lo w tem p eratu re su p e rco n d u c to rs th e ratio
A /£ «
1. F or e x a m p le , in pure a lu m in u m w ith Tt ~ 1 .2 K ,
£ (0 ) = 1600nm .
A (0 ) = 5 0 n m w h ile
T h is im p lie s that o n e can n o lo n g e r a p p ly the lo c a l L o n d o n
electro d y n a m ics. Instead, the B C S theory su g g e sts that the fie ld s m ust b e averaged
o v e r the c o h e r e n c e len gth . T h e ex p r e ssio n for the current d en sity then tak es the
n o n -lo cal form
:_____ f R [ R ^
J ( r ) = - e . n v aL22t (- D J I R l l t ‘R A
, r )l"
4
w h ere k L rep resen ts th e L on d on p en etration d ep th .
(1-20)
T h e th eory w o rk s w e ll for
c o n v en tio n a l lo w tem perature superconductors.
In th e c a s e
A = 1 6 0 0 A - 8 0 0 0 A , i.e .,
o f ty p e -II
A »
H TSC
o x id e s
£
~
5 A -3 0 A
and
T h is im p lie s that m a g n e tic fie ld p en etrates
o v er a d istan ce m uch larger than the len gth o v er w h ich the fie ld averagin g is carried
o u t in E q .( 1 .2 0 ) .
T h e r e fo r e , the H T S C
o x id e s ca n
b e tr ea te d as lo c a l
su p erco n d u cto rs and th e e le c tr o d y n a m ic s can b e ad eq u a tely d e sc r ib e d u s in g the
L ond on eq u ation s.
14
1.6
Conductivity in the Superconducting State — Two
- Fluid Model
E x p e rim en ta lly [24] the tem perature d ep en d e n c e o f A for T < T c is
sh o w n to fit rather w e ll to the form
(
1.2 1 )
U sin g Eq. (1 .2 1 ) in Eq. (1 .1 8 ) and a ssu m in g all the tem perature d ep en d e n c e to be
c o m in g from the d en sity o f su percon d ucting electro n s n s , on e ob tains
( 1 .2 2 )
w h er e
n 0 re p r e se n ts th e total n u m b er o f e le c tr o n s .
S in c e
n 0 is fix e d the
tem perature d ep en d en ce o f the norm al electron s can be obtained from Eq. (1 .2 2 ) as
~ n0
nS —n0
( 1 .2 3 )
\ T‘ c /
T h e refo re at any fin ite n o n -z e r o tem p eratu re T < T c th e carriers are both norm al
ele c tr o n s and su p e rco n d u c tin g e le c tr o n s and h en ce n am e - tw o -flu id m o d el. T h e
current d en sity in su ch a ca se can be written as the sum o f the current d en sity due to
the norm al carriers and that due to the su percon d u ctin g electron s. From E qs. (1 .1 2 )
and (1 .1 8 ) the supercurrent d en sity is ob tain ed as
(1 .2 4 )
15
and from O h m ’s law the norm al state current d en sity is
J„ = OAT) E
(1.25)
w h ere O^iT) is the tem perature d ep en d en t d c co n d u c tiv ity o f the norm al ch an n el.
T h erefore, the total current d en sity is
J —J s + J N-
T + o - ,( r ) E .
( 1. 2 6 )
S o that the con d u ctiv ity o o f a superconductor is a co m p lex quantity
a = <7, - i o ,
( 1 .2 7 )
w h e r e th e c o n d u c t iv it y o f th e n o rm a l c h a n n e l,
✓ \4
T
d ep en d en ce <7, = a N
and
<7,, h a s th e te m p e r a tu r e
( 1. 2 8 )
a ,(D =
H qcdX 2( T )
16
1.7
L andau-G inzburg Theory
T he L on d on m od el d o es not co n sid er the field or current variation o f
the d en sity o f su p e relec tro n s ns.
T o take into a cc o u n t the e f fe c t o f extern al
param eters su ch as fie ld on ns, on e has u se the L a n d a u -G in zb u rg [2 4 ]
m od el
w h ich is b ased on ex p a n d in g the free en ergy in term s o f the su p erco n d u ctin g order
param eter d efin ed as
^ ( r ) = |v ^ ,( r ) |^ <r»
( 1 .2 9 )
w here 6 ( r ) is related to the p h ase o f the supercurrent and the am p litu d e g iv e s the
num ber o f su p ercon d u ctin g elec tro n s or C o o p er pairs, i.e., yzs(r )y r j( r ) = n y( r ) .
M in im iz in g the free en erg y w ith resp ect to the order param eter and
the v ector potential A on e ob tain s the w ell know n L andau-G inzburg eq u ation s
oe\f/s + P \ y / S\2
+ — (iAV - 2 e A ) 2 y/s = 0 ,
2m
J = c u r l h = — [ yf*s ( - i h V - 2 e A . ) y / s 1
m 1
1
(1 .3 0 a )
( 1,3 0 b )
w h ere a and /? are param eters that are related to the characteristic q uan tities such
as therm odynam ic critical field H ( , penetration depth A and the c o h e r e n c e len gth £
E qs. (1 ,3 0 ) are c o u p le d e q u a tio n s w h ere the first o n e y ie ld s the order param eter
w h ile the se c o n d o n e g iv e s the supercurrent d en sity in the su p ercon d u ctor. S o lv in g
th ese tw o eq u ation s on e co u ld calcu late the H , , A a n d £ . A n im portant relation sh ip
b etw een th ese three quantities, ob tain ed from the L andau-G inzburg m o d el, is
17
d)
H ( T ) k { T ) B { T ) = con stan t = ------- SL1=2 ff /i0 V 2
( 1 .3 1 )
w here <J>() = 2 . 0 5 x 10_lsT m 2 .
1.8
Type-I and Type-II Superconductors
A ll the p rop erties d escrib ed in se c tio n s 1-5 are true o f both type-1
and type-11 m aterials. H o w ev e r, there a lso ex ist sig n ifica n t d ifferen ces b etw e en the
tw o . T h e prim ary d iffe r e n c e is that w h ile the m a g n etiz a tio n o f th e ty p e-I drops
sh arp ly at a certain cr itica l fie ld
(F ig . 1.1 a) that o f typ e-II (F ig . 1.1b ) s h o w s a
gradual d ecrease as the applied field is increased b eyon d the lo w er critical field Hc | ,
is n o n -zer o in the fie ld region H c i < H < H c 2 and red u ces to zero o n ly b ey o n d the
u pper critical fie ld H c 2
In other w ord s in type-11 m aterial, fo r H > H c 1, the field
enters the sp ecim en as vortices. T he p resen ce o f vortices or flu x o n s in th ese type-11
sa m p les at high ap p lied fie ld s h ave b een co n firm ed by m any ex p er im en ts [25] and
are k n o w n to d ra m a tica lly in flu e n c e the p rop erties o f the m aterial.
O n e o f the
s u c c e s s e s o f th e L an d au -G in zb u rg m o d el w a s the p red iction o f the e x is te n c e o f
type-11 su p ercon d u ctors.
T he p h y sica l reason for the form ation o f v o rtices in typ e-II m aterial
can be u n d erstood from sim p le en ergy argum ents as fo llo w s. C on sid er a boundary
b etw een a su p ercon d u ctin g and a norm al m etal w ith the applied field H e parallel to
the b oun d ary. T h e lo ss in m a g n etic en erg y due to penetration o f the a p p lied field
into the su percon d uctor is / i 0
w h ile the system gain s in con d en sation en ergy
on g o in g from the boundary into the su p ercon d u ctor by / t () H ^ / 2 . C o n seq u e n tly
a su rface en er g y .
18
E s = [ Z - X ) f i 0 H t: / 2
a rises.
If £ > A ,
(
1.3 2 )
Es > 0 and th e su p erco n d u cto r w ill tend to rem ain sp a tia lly
h o m o g e n e o u s, w h ile if £ < A , as is the c a s e w ith the typ e-II m a teria ls, then the
su p ercon d u ctor w ill tend to break up into som e region s w h ich are norm al and som e
that are su p ercon d u ctin g. T h e ap plied field is restricted to th ese norm al region s and
form v o r tic e s.
T o ob tain the m a x im a l n um ber o f d o m a in s ea c h v o rtex sh o u ld
in c lu d e
s m a lle s t
th e
u n it
of
m a g n e t ic
f lu x
c a lle d
th e
flu x o n ,
<D0 = h j l e = 2 .0 5 x 10 _IST - m 2.
A d e s c r ip tio n o f a s in g le v o r te x
s u p e r c o n d u c to r s w h e r e k » B , ,
ca n
in the c a s e o f th e ty p e-II
be o b ta in e d by s o lv in g
fo r th e fie ld
d istrib u tion u sin g th e L an d au -G in zb u rg eq u a tio n s. T h e structure o f the vortex is
sh o w n s c h e m a tic a lly in F ig . 1.4 [2 6 ]. T h e core o f the vortex is n orm al and has a
radius o f £ (ig n o rin g th e a n iso tro p y , for the p resent d is c u s sio n ).
A supercurrent
flo w s in the su p erco n d u ctin g region surrounding the core. T h e field is m axim u m at
the cen ter o f the core and d eca y s o v e r a len gth sca le g iv en by the penetration depth
A . A s see n in F ig. 1.4c, the su p erco n d u ctin g order param eter y/s g o e s to zero at
the cen ter o f the core and recovers its full value at a distance % from the core center.
19
(a)
h
(b)
r
(c)
F ig . 1.4.
C r o ss s e c tio n o f an iso la te d vortex in a typ e-II su p erco n d u cto r (from
R e f [2 6 ]). (a ) current (b ) fie ld (c) pair d en sity.
20
For the typ e-II m aterials, on e can m ake a straigh forw ard estim ation
o f the upp er and lo w er critical fie ld s from the v a lu e s o f £ and A ,
as fo llo w s .
W h en the ap p lied field reach es H C1 field b egin s to penetrate the sp ecim en . In other
w o r d s, it b c o m e s e n e r g e tic a lly fa v o r a b le to h ave a flu x o n in sid e and sin c e the
average field is H c ) and is spread o v er the length A , the flu x sh o u ld be
<D0 =
T h erefore,
H , =
ttA
V oH c].
t^ ° , .
W o*
( 1 .3 3 )
A s th e a p p lied fie ld is in c rea se d , the d e n sity o f flu x o n s in the
sp ecim e n in crea ses and c lo s e to H c 2 the d istance b etw een the v o rtices ap proach es
the c o h er en ce len gth .
T he area o cu u p ied by each vortex is then ro u g h ly n%2 and
o n e obtains
H lo = ~ “ T T Wo£~
d -3 4 )
A m ore p recise ex p ressio n for H lI can be obtained from sim p le en ergy argum ents is
- ^ Ml n n* Hc l= —
Ho*
4
and w e n o tic e that for X » £ , , B C 1 is sm a ll.
( 1 .3 5 )
U s in g E q . ( 1 .3 1 ) , w e fin d an
interesting relationship b etw een the tw o critical field s and the th erm od ynam ic field ,
H clH i 2 = H > 4 .
s
21
<! -3 6 )
S in c e H c d o e s not vary very m uch in typ e-II su p erco n d u cto rs and
sin c e in the h ig h -T c o x id e s H c 2 is as m uch as 100 T due to the sm all co h er en ce
le n g th , H c i in th e se m a teria ls can be e x tr e m e ly sm a ll.
In o th er w o r d s, flu x
penetration can occu r for m oderately large field s.
1.9
Pinning in Type-II Superconductors
T h e H T S C o x id e s are w e ll k n ow n to h ave variou s d e fe c ts su ch as
grain b ou n d aries, v o id s, re g io n s o f d ep elete d o x y g e n co n ten t etc. T h e se d e fe c ts
act as p in n in g cen ters for the v o rtices b eca u se it is e n er g etica lly fa vorab le for the
su p erco n d u ctor to p la c e a vortex w h ich has a norm al core on the d e fe c t site. T he
degree o f p in n ing has a sign ifican t e ffe c t on the transport and m agn etic properties o f
the m a terial.
In m a teria ls w h ere p in n in g is u nim p ortant, th e v o r tic e s are free to
m o v e in and ou t o f the sa m p le as field is varied through H e 1, and th ese are know n
as so ft su percon d uctors. W h en p in n ing b eco m es sig n ifica n t the flu x o n s can rem ain
trapped at the p in n ing site and not readily m o v e with varying field; su ch a m aterial is
term ed a hard su perconductor.
S u p p o se an extern al current w ith a current d en sity J is ap p lied to a
su p e rco n d u c tin g sa m p le w h ich has th ese v o rtic es in it.
A L oren tz fo r ce d en sity
f = I x <t>|( w ill be ex p erien ced by the flu x o n s due to the applied current. T he effect
o f the L oren tz force is to m o v e the flu x o n , ca u sin g d issip ation . H o w ev e r, if p inning
is su ffic ie n tly strong then it is p o ssib le for the flu xon to rem ain stationary and there
w ill be n o d issip a tio n . T h e p in n in g force d en sity is u ltim a tely d eterm in ed by the
d efe ct structure o f the sam p le. W h en a b alan ce b etw een the L oren tz fo rce and the
pinning force is ach iev ed the sam p le is said to be in the critical state and this d efin es
the critical current density J c .
T o d escrib e a su p ercon d u ctor w ith strong p in n in g, C. P. B ean [27]
p ro p o sed the critical state m od el. T he flu x d en sity is assu m ed to lead to a field B
in the sa m p le w h ich sa tisfies the relation
V x B = ^ 0J t
for any a p p lied fie ld v a lu e .
( 1. 37)
In oth er w ord s, the B ea n m od el su p p o se s that any
ap p lied fie ld a lw a y s p rod u ces the critical current d en sity. Furtherm ore, J L., w h ich
in p rin cip le is a s lo w ly v aryin g fu n ctio n o f fie ld , is c o n sid e r e d h ere to be field
independent.
W e c h o o se the sim p le ca se o f a su percon d ucting slab to illustrate the
critical state b eh avior. L et d b e the th ick n ess o f the slab and a field H z b e ap plied
parallel to its su rface as sh o w n in F ig. 1.5a.
W h en the a p p lied fie ld ju st e x c e e d s
Hc i, field penetrates the slab to a th ick n ess determ ined by the critical con d ition
~
= l
ax
T he d ista n c e to w h ich the field p en etrates,
jc0 ,
(1 .3 8 )
is g iv e n b y
jc0
= — .
•A
W h en the
a p p lied field reach es the penetration field H \ the field p ro file is as sh o w n in Fig
1.5b, and the field rea ch es the cen ter o f the slab, i.e ., H * =
% •
A s the ap p lied
field is in creased b ey o n d H \ the field at the cen ter in crea ses, h o w e v e r , the slo p e
rem ains th e sam e and the average currrent d en sity has the sam e distribution. W h en
the ap p lied field is redu ced, so m e o f the flu x has to lea v e the sam p le so as to satisfy
the b ou n d ary c o n d itio n at the su rface o f the slab. S in c e flu x is stro n g ly p in n ed ,
o n ly th o se flu x o n s c lo s e to the su rface le a v e the sam p le w h ile in the in terior the
23
sa m p le tries to m ain tain the flu x d e n sity as b efore (s e e F ig. 1.5d).
O n e can se e ,
th erefore, that ev e n w h en the extern al field is brought to zero, co n sid era b le am ount
o f flu x w ill rem ain trapped w ith in the sa m p le.
O ne w o u ld h ave to re v erse the
ex tern a lly a p p lied field to d rive out the trapped flu x. From th ese resu lts, it is clea r
that a p p ly in g a tim e v a ry in g m a g n e tic fie ld to a su p e rco n d u c tin g m aterial w ith
stron g p in n in g w ill lea d to h y ste r e sis. B e a n ’s m o d el y ie ld s o n e a straigh tforw ard
w ay to estim ate the critical current d en sity o f a sp ecim en o n ce the d im en sio n s o f the
sa m p le are k n o w n . H o w ev e r, o n e n eed s to k eep in m in d that in a num ber o f w a y s
th e a b o v e is a s im p lis tic treatm en t.
In p articu lar, w e h a v e ig n o r ed th e fie ld
d ep en d en ce o f J c and ch o se n the sim p le geom etry o f a slab.
24
C-l
T!
Tc
o
X
o
a
V
t
25
the critical state model for a superconducting
t
Fig. 1.5. The average flux density as obtained from
t
slab
1.10 Anisotropy in HTSC Oxides
T h e p erovsk ite structure o f m ost o f the rare earth based and bism uth
b ased co p p er o x id e s results in unit c e lls that h ave d ifferen t lattice co n sta n ts a lon g
the three axes. A sch em atic o f an unit c e ll o f Y B a 2 C u 3 0 7 supercon d uctor is sh ow n
in F ig . 1.6.
T h e d im e n s io n s o f the u nit c e ll are a p p r o x im a te ly
- 0 .4 n m alon g the c -a x is and the a- or b- a x es resp ectiv ely .
~ 1.2nm and
T h is an isotrop y o f the
u n it c e ll structure is r e fle c te d in th e a n iso tro p y o f m an y o f th e fu n d a m en ta l
param eters o f the m aterial su ch as the penetration depth, co h er en ce len gth and the
e n erg y gap. C o n se q u e n tly , m any o f th e p rop reties o f th ese m aterials, su ch as the
c o n d u c tiv ity , are a ls o d ir ectio n d ep en d en t.
T a b le 1.1 [2 8 ] lists
th e e stim a ted
penetration dep th s and co h eren ce len gth s for three H T S C o x id es.
Table 1.1.
Estimated penetration depths A. and coherence lengths ^
along the unit cell axes (j = a,b,c).
Material
A-u.b
K
(nm )
(nm )
m)
(nm )
Y !B a 2C u 10 7
15 0
600
1.5
0 .4
B i 2S r 2C a 2C u ,O l0
200
10 0 0
1.3
0 .2
80
400
3 .5
0 .7
L a 2_ ,s rxC u 4
£
26
•
Cu
"Cu 0 c ha ins ”
c
F ig . 1.6.
S c h e m a tic d ia g ra m o f th e u nit c e ll o f Y B a 2 C u 3 C>7 (Y B C O ).
The
o rth o rh o m b ic structure h as unit c e ll d im e n s io n s a = 3 .8 8 A , b = 3 .8 4 A , and c =
1 1 .6 3 A w ith a c e ll v o lu m e ~ 173 A 3 .
27
1.11 Josephson Junction
C o n sid e r a san d w ich structure o f the form norm al m etal-in su latorn orm al m etal w h er e the in su la to r th ic k n e ss is s u ffic ie n tly thin ( - 1 0 - 2 0 A ) .
Q uantum m ech a n ics predicts that there e x ists a certain probablity for an electron to
tu n nel th rough the in su la tin g barrier.
A t lo w v o lta g e s , the I-V cu rve o f such a
n o rm a l-m eta l tu n nel ju n c tio n w o u ld be o h m ic [2 4 ], i.e ., the current is d ir ectly
p ro p o rtion al to the a p p lied v o lta g e .
H o w e v e r , if o n e o f the e le c tr o d e s o f th is
ju n c tio n w ere to be su p erco n d u cto r, the form o f the I-V cu rv e w o u ld b e q uite
d iffer en t (s e e F ig . 1.7).
W e s e e that at z e r o -v o lta g e n o current flo w s until the
applied voltage reaches a characteristic value V , the gap v o lta g e, w h ich is g iv en by
w h ere 2 A , the en erg y g ap o f the su p ercon d u ctor, is the en erg y required to break
the C o o p er pair.
In 1 9 6 2 , B rian J o sep h so n p red icted that if tw o su p e rco n d u c to rs
w ere sep arated by a s u ffic ie n tly thin in su la tin g layer, then o n e co u ld e x p e c t the
tunneling o f the C oop er pairs through the potential barrier w ith ou t g iv in g rise to any
v o lta g e [2 4 ].
In o th er w o r d s, a su percu rren t w o u ld p a ss th rou gh the ju n ctio n
w ith ou t d issip ation . T h is has c o m e to be know n as the dc J osep h son e ffe c t and the
ju n c tio n s as J o se p h so n ju n c tio n s .
F ig. 1.8
sh o w s th e re su ltin g I-V cu rve; the
J o se p h so n current is rep resen ted by the current at ze ro v o lta g e .
T h e p h y sica l
reason in g for this is that the w a v efu n ctio n s d escrib in g the C oop er pair in
28
F ig . 1.7
Id ea lized form o f the i-v cu rve for an S IN ju n ctio n [2 4 ], T h e so lid lin e
rep resen ts the ju n c tio n b eh a v io r b e lo w th e su p e r c o n d u c tin g critica l tem perature
(7 ^ ) w h ile the dashed lin e is for T > T r .
N orm al E lectron
T u n n e lin g
C o o p e r Pair
T u n n elin g
2A
2A
F ig . 1.8
C u r r e n t-v o lta g e c h a r a c te r is tic s [2 4 ] fo r an u n sh u n te d J o s e p h s o n
ju n ctio n . T h e current at zero v o lta g e is the J o sep h son current.
29
/
/
Superconductor
/
Insulator
/
©
Superconductor
F ig . 1.9a.
S c h e m a t ic
of
a
cu rren t
su p ercon d u ctor (S IS ) tunnel ju n ction .
30
d r iv e n
s u p e r c o n d u c t o r - in s u la t o r -
in each electro d e are phase lock ed togeth er so that the phase co h eren ce b etw een the
tw o w a v e fu n c tio n s e x te n d s o v er both ele c tr o d e s. A p h ase d iffe r e n c e acro ss the
ju n c tio n c a u s e s the flo w o f a su percu rren t.
T he
w e ll k n o w n [2 4 ] J o sep h so n
rela tio n b e tw e e n th e cu rren t th rou gh th e ju n c tio n 1, th e cu rren t d e n sity b e in g
a ssu m ed to be u niform o v er the ju n ctio n area, and the p h a se d iffe r e n c e (p, in the
a b sen ce o f any external m agn etic field is
/ = I( sin (p
w here
<p = ( 0 , - 0 2)
and
( 1 .4 0 )
0, d e n o te s th e p h a se o f the c o r r e s p o n d in g ord er
p aram eter, y/i = \ y / l\e'v>' in su p e r c o n d u c to r i.
Eq. ( 1 .4 0 ) is k n o w n a s the dc
J osep h son relation. A m ore general exp ression for the v o lta g e-p h a se relation can be
fo u n d b y c h o o s in g a g a u g e su ch that the v ec to r p oten tial v a n ish e s, i.e ., o n e can
c h o o se a vector potential A (r ,r ) = - V ^ ( r , r ) so that A ' = A +
= 0
T h e g a u g e-
invariant phase d ifferen ce is then g iv en by
2
<P( y , z, t ) = 0, ( v, z , t )
- 0 2 (y, z, t )- ~
f A ( r,t)
0
•d l
(141)
,
D iffe r e n tia tin g E q. ( 1 .4 1 ), th e relation b e tw e e n th e v o lta g e d rop , v , a cr o ss the
ju n ction and the phase <p are relatedby
d*P
2 jt
- r = — v.
dt
<D0
,,
(1 .4 2 )
E qs. ( 1 .4 1 ) and (1 .4 2 ) are the fu n d am en tal e q u a tio n s g o v e r n in g the b eh a v io r o f
J o sep h so n ju n c tio n s.
31
T he m ax im u m d c current that can be p a ssed th rou gh a ju n ctio n at
zero v o lta g e is k n ow n as the Josep h son critical current Ic . T he m ax im u m current is
tem perature d ep en d en t and related to the su p ercon d u ctin g en erg y gap A (T ).
For
th e c a s e o f an S IS ju n ctio n m ad e o f tw o id en tical su p ercon d u ctors th e tem perature
deperature d ep en d en ce is g iv en by the A m b egaok ar-B aratoff [29] relation
I (T)= ^ ^ -ta n h ^ ^
2eRN
2kT
( 1 .4 3 )
w here R n is the ju n ction tunnel resistance.
It is w e ll k n ow n [30] that ap p lication o f a m a g n etic fie ld d am p s the
c r itic a l cu rren t o f a J o s e p h s o n ju n c tio n u n d e r g o e s d a m p in g , an d th e fie ld
d ep en d en ce resem b les the diffraction pattern produced by a sin g le -slit. W h ereas for
an o p tical d iffra ctio n pattern, it is the ratio o f slit d im e n sio n to the w a v e len gth o f
th e lig h t that c o n tr o ls the p o sitio n o f th e m a x im a , for a J o se p h so n ju n c tio n the
critica l current m a x im a are d eterm in ed by the area o f the ju n c tio n and n um ber o f
v o rtic es in it. C o n sid e r the sim p le c a se o f a short J o sep h so n ju n c tio n ; i.e. (i) the
m a g n e tic fie ld p ro d u c ed b y th e cu rren ts th rou gh th e ju n c tio n is n e g lig ib le in
co m p a riso n w ith the ex tern a lly ap p lied fie ld , and (ii) th e len g th and w id th o f the
ju n ctio n are sm a ll co m p a red to a ch aracteristic len g th , k n o w n as the J o sep h so n
penetration depth. T he cro ss sectio n o f such a Josep h son ju n ctio n is
32
h
v
F ig . 1.9b.
C r o ss se c tio n o f a J o sep h so n tunnnel ju n ctio n m ad e o f tw o
su p e rco n d u c tin g e le c tr o d e s o f th ick n e ss b, and b 2 w ith penetration d ep th s
X] and
r e sp e c tiv e ly and an in su latin g barrier o f th ick n ess 2a. T h e
current flo w s a cro ss the ju n ctio n in the x-d irection and the ap p lied fie ld
is a lo n g the y -a x is.
33
sh o w n in Fig. 1.9b. Let b | and b2 be the th ic k n e sse s o f the tw o su p erco n d u ctin g
electro d es with A[ and A 2 as their resp ective L on d on penetration d ep th s and 2a be
the th ick n ess o f the insu latin g barrier. Let d be the length o f the ju n ctio n in the z d ir e c tio n and w th e w id th in the
c o n sid e r in g d » 2 a and w » 2 a .
a p p lied a lo n g the n eg a tiv e
v -d ir e c tio n .
E d g e e f f e c t s are ig n o r e d by
W e w ill furtherm ore su p p o se that th e current is
jc-d irection
and B = fi0y is the e x te r n a lly a p p lied dc
fie ld . T h e m a g n etic fie ld d e c a y s e x p o n e n tia lly into ea ch o f the su p e rco n d u c tin g
e le c tr o d e s lead in g to an e ffe c tiv e ju n ctio n th ick n e ss g iv e n by heJf = A, + A 2 + 2 a .
U sin g Eq. (1 .4 3 ) o n e ob tains for the phase ch an ge across the ju n ction
(1 .4 5 )
( p ( z ) ~ ( p ( 0 ) = — B0heJfz
U sin g the cu rren t-p h ase relation a lo n g w ith E q. (1 .4 5 ) and in tegratin g th e current
d en sity o v er the ju n ction area, on e obtains for the total current through the ju n ction
( 1 .4 6 )
w here
= B0hrffd represen ts the flu x through the ju n ctio n . From Eq. { 1 .4 6 ) it is
e v id e n t that th e m a x im u m
sin (p(0 ) = ±1 and therefore
su p ercu rren t f lo w s th ro u g h th e ju n c tio n
w h en
X
3
e
"3
"5Q*
C3
4>
3
"O
1)
CN
Ic(<P)/Ic(0 )
CJ
1)
o
e
O
E
w
3
cx
c
.2
&
w
i
CO
I
^aj
o
j=
s
a
«ba
a.
o
w>
lE
35
Eq. (1 .4 7 ) y ie ld s the w e ll-k n o w n F rau n hofer m a g n etic d iffra ctio n pattern o f the
ju n ctio n critical current (s e e F ig. 1.10). S everal m o d els h ave b een put forw ard to
ex p la in th e b eh a v io r o f critical current in granular H T S C m aterials b a sed on the
b eh a v io r o f J o sep h so n ju n c tio n s [3 1 , 32].
1.12
Resistively Shunted Josephson Junction (RSJ)
S o far the d isc u ssio n has in v o lv e d ideal J o sep h so n ju n ctio n s
at T = 0 . For any tem perature T c > T > 0 on e also has to take into accou n t quasiparticle
tu n n elin g . T o d o th is o n e u ses a resista n ce in p arallel w ith the ju n c tio n and the
resultan t c irc u it is term ed the r e sis tiv e ly sh u n ted J o sep h so n ju n c tio n (R S J ).
A
ty p ica l RSJ circu it is sh o w n in F ig. 1.11. In p rin cip le, o n e sh o u ld a lso c o n sid e r the
e ffe c t o f cap acitan ce due to the parallel plate structure o f the ju n ctio n . T h e RSJ can
eith er be d riven by a v o lta g e sou rce or a current sou rce and for the p resen t w e are
in terested in the current d riven ju n ctio n . T h e w e ll k n ow n [2 9 ] R SJ eq u a tio n , in
such a ca se , is then ob ta in ed by u sin g K ir c h o ffs current law and the v o lta g e-p h a se
relation o f the JJ as
+
+
R 2 jt d t
( 1 .4 7 )
2 n dt2
w h ere the left hand sid e in the total current through the circuit. T he first term on the
right hand sid e is the J o se p h so n ju n ctio n current, the se c o n d term is the current
th rough th e r e sistiv e branch w h ile th e third is that through the ca p a c ita tiv e branch.
T h is resultant n o n -lin ea r eq u a tio n is u sed to understand the b eh a v io r o f th e RSJ.
S ev era l attem pts to m o d el the b eh avior o f granular su percon d u ctors b ased on R SJs
are ev id en t in the literature [33, 34],
36
I ©
F ig . 1.11.
lc sin<pX
A current driven re sistiv e ly shunted J osep h son ju n ctio n circuit
w h ere R is the resistan ce o f the ju n ction and C is the cap acitan ce c o m in g
from the parallel plate structure o f the ju n ction .
current in the J osep h son ju n ction .
37
Ic simp represents the
1.13 Surface Impedance
In the ca se o f a bulk su p ercon d u ctor the su rface im p ed a n ce is g iv en
by the w e ll k n ow n [35] equation
1/2
( 1 .4 8 )
a
w h ere a in the su p erco n d u ctin g state is g iv e n by Eq. (1 .2 7 ). T h e real part o f E q.
(1 .4 8 )
y ie ld s the surface resistan ce w h ich is the lo ss and is g iv e n by
1/2
1/2
tT|2 + a 2 - a 2
_2 , _2
( 1 .4 9 )
a , + <7,
w h ere cr, is related to the skin depth S by
f
2
V' 2
S=
(1 .5 0 )
CUtT,1
and <7, is g iv e n by Eq. (1 .2 8 ),
1
0-, =
(1 .2 8 )
t l 0Q)X2
N o r m a liz in g Eq. (1 .4 9 ) to the the lo ss in the norm al state
(
V /2
V
y
(1 .5 1 )
w h ere o N is the con d u ctivity in the normal state w e write
38
Eq. ( 1 .5 2 ) g iv e s the p o w e r ab osrb ed by the su p erco n d u cto r.
From E q s. (1 .2 7 ),
( 1 .2 8 ), (1 .5 0 ) and (1 .5 2 ) w e see that in the norm al state, sin c e A is in fin ite , the
a b so rp tio n is c o n tr o lle d by th e sk in d ep th
5.
H o w e v e r , at th e o n s e t o f
su p e rco n d u c tiv ity , a rapid d ecrease o f A w ith d ecr ea sin g T b rings about a sharp
red u ction in the p o w er ab sorb ed by the sam p le.
It is u sefu l to w rite Eq. (1 .5 2 ) in
term s o f the tw o fundam ental length sca les A and 8 as
( 1 .5 3 )
|
( l + ( V 2 A / 5 ) 4)
Chapter II
Method
2.1. Sam ples
T h e v a riety o f sa m p le s in c lu d e d Y B a 2 C u 3 0 7 -g (Y B C O ) and
B i i ,6 P b o .3 S b o . | S r 2 C a 2 C u 3 0 1 0 (B S C C O ) p o w d ers
o f d iffe r e n t grain s iz e s ,
Y B C O a g g lo m e r a te s and an a lig n e d Y B C O p o w d er. T a b le. 2.1 lists th e sa m p les
and their n om in al grain sizes. For com p arison [1 7 ] o n e a lso lo o k e d at s o m e thin
film s o f Y B C O , a san d w ich type structure o f P rB C O /Y B C O /P rB C O on a L a A 1 0 3
su bstrate, sin g le cr y sta ls o f Y B C O and E rB C O [3 6 ] as w e ll as a tw in n ed Y B C O
s in g le crystal.
S a m p le N o .
I w a s o b ta in e d fr o m A K Z O w h e r e th e y u se a
p rop reitary te c h n iq u e w h ile N o . 2
w a s lo c a lly p rep ared by a h ea t-g rin d -h ea t
m eth o d [3 7 ]. In this m eth o d o n e starts w ith o x id e p o w d ers - Y 2 O 3 , B a C 0 3 and
C u O o f h ig h purity. T h e se p o w d ers are m ix ed in the correct sto ic h io m e tr ic ratio
and th en h ea ted in air fo r fou r hours at 9 5 0 °C . T h e m ixtu re is c o o le d to room
tem p era tu re g ro u n d and reh ea ted to 9 5 0 °C in o x y g e n a tm o sp h ere .
m aterial is th en s lo w ly c o o le d to room tem p eratu re and
T h e fin a l
g ro u n d to m ak e the
p o w d er sa m p le s.
N . D . S p e n c e r o f W . R. G ra ce and C o . k in d ly p r o v id e d
sa m p le s N o . 3, 4 , 5 and 9 w h ich w ere grow n by co -p r ecip ita tio n m eth o d [3 8 ],
i.e ., the sa m p le s w ere p recip itated from a so lu tio n co n ta in in g the appropriate
m etal nitrates or carbonates in correct stoich iom etric quantities. T he preparation
o f B S C C O (sa m p le 9 ) for e x a m p le , in v o lv e d co p recip ita tio n o f b ism u th , lead,
40
stro n tiu m , c a lc iu m and co p p er io n s as carb on ates o n to fin e ly d iv id e d S b 2 0 s
n u clei. A fter filtration, d rying and h eatin g in air at 5 40°C for 5 hours fo llo w e d
by heat treatm ent at 8 00°C for 12 hours, the black p ow d ers w ere then p elletize d
u sin g a 2 0 ,0 0 0 lb p ress and sintered in air for 6 0 hours at 8 5 0 °C tw ic e . Each
tim e c o o l d o w n w a s co n tr o lle d at the s lo w rate o f < T C /m in u te to 3 5 0 °C
fo llo w e d by a rapid c o o lin g to room tem perature. X -ray a n a ly sis o f sa m p le 9
[3 9 ] in d ica ted a s in g le p h a se c o m p o u n d and su sc e p tib ility stu d ie s sh o w e d a
nearly 100% su p ercon d u ctin g fraction in th e p ow d ered m aterial w ith a 7" = 106
K.
S a m p le 6 is an a lig n e d p o w d er c o n s is tin g o f 2 x 2 x 1,5 p m
Y B C O g r a in s w ith c - a x is o r ie n ta tio n e m b e d d e d in e p o x y .
A d e ta ile d
d escrip tio n o f th is c -a x is alig n ed sam p le can b e fou n d in R ef. 3 9 . T h e sa m p le
w a s prep ared b y settin g th e grain s in a 5 -m in u te e p o x y and a p p ly in g a large
field o f 8 T at room tem perature.
S a m p le s 7 and 8 in the form o f p ellets w ere prepared by N ath e t al.
[4 0 ] u sin g the co n v en tio n a l heat-grind-heat tech n iq u e d escrib ed above.
T h e av era g e grain s iz e for ea c h sa m p le w a s d eterm in ed as fo llo w s .
U sin g an o p tica l m icr o sco p e, the calip er siz e o f a num ber o f grains (1 0 0 to 2 0 0 ) o f
a g iv e n sa m p le w a s m easu red and a h istogram ob ta in ed [3 9 ]. T h e a v er a g e grain
s iz e (d iam eter - 2 R ) is equal to the m od e o f a sim p le gaussian distribution. A typical
grain s iz e d istrib u tion for the 10 f i m Y B C O p o w d er (sa m p le 3 ) is sh o w n in F ig.
2 . 1.
41
c
'0
SJ)
DXj
s
>
“O
5
o
Cl
sc
C
V //////////A
CO CO
in
o
o
a
’s
u
*3
a.
0
CM
E
2CJi
c
i/3
1
10
CM
oCNJ
ID
o
lO
cj
du
iZ
ja q u jn N
42
10|im
u
size is roughly
73
(1 division = 1.9 fim ).
o
u
ca
>
Table U .l .
List of samples used in the study and their
respective grain diameters.
S a m p le N o .
T ype
N om in al diam eter (p m )
1
Y B C O Pw dr (A K Z O )
2
2
Y B C O Pw dr
6
3
Y B C O Pw dr
10
4
Y B C O agglom erate
50
5
Y B C O agglom erate
88
6
Y B C O aligned Pwdr
2
7
Y B C O P ellet 1
-
8
Y B C O Pellet 2
-
9
B S C C O P w dr
5
43
T h e m icron s iz e p ow d ers and a g g lom erates w ere h eld on thin (0 .1 6
m m ) quartz p lates u sin g dilute G E 7 0 3 1 cem en t. T he d im en sion o f the quartz plate
w a s ro u g h ly I m m x 1m m at 10 G H z and m uch sm aller at h igh er fr eq u e n c ie s. In
e a c h c a s e th e p o w d e r w a s m ix e d in th e ce m e n t and the m ix tu re p ain ted o n to a
quartz plate. T h e g lu e on drying left behind a uniform thin coatin g o f the sam p le on
the quartz plate. T o obtain an idea o f the e ffe c tiv e fillin g fraction o f the sam p le on
the quartz p late, several sa m p les w ere carefu lly w e ig h ed and their e ffe c tiv e d en sity
w a s e stim a te d to b e around 7% to 14% o f the bulk d en sity (a g g r eg a te o f atom ic
m a sse s/u n it c e ll v o lu m e ).
T h is y ie ld e d an e f fe c tiv e fillin g factor o f about 10%.
C h a n g in g the fillin g fraction d id not a ffect the ob servation s sig n ific a n tly and h en ce
led o n e to c o n c lu d e that fo r th e d ilu tio n u se d , in terp a rticle in tera c tio n w a s
n eg lig ib le.
2.2
M icrowave Spectrom eter
T h e freq u en cy range u sed in this study w a s 2 G H z - 6 0 G H z. T o
c o v e r this range fou r d ifferen t E S R sp ectrom eters w ere u sed . T h e 2 G H z -6 G H z
sp ectro m eter u sed a c o a x ia l c a b le term in ated by a cy lin d r ic a l c a v ity . A d eta iled
d escrip tio n o f th is sp ectrom eter can be fou n d in R ef. 4 2 . R ectan gu lar w a v e g u id e s
and c a v itie s w ere u se d at all oth er fr e q u e n c ie s.
T he 10 G H z c a v ity ( o f rough
d im en sio n 2 .5 cm x 2 .2 cm x 1.1 cm ) is a T E ) o 1 w h ile the h igh er o n e s are TE ] On
w ith n = 4 or 5. T h e c a v itie s are m ostly m ad e o f Cu or brass e x c e p t th e o n e at 2 G H z
w h ic h w a s m ad e o f a lu m in u m . T h e Q o f the rectan gu lar c a v ity is ro u g h ly 2 0 0 0
w h ile the cy lin d r ic a l ca v ity has a Q o f a fe w hundred.
A d eta iled d escrip tio n o f
th ese can be fou n d in p reviou s reports from this laboratory [4 3 ]. F igu re 2 .2 sh o w s
the sch em a tic o f the m icr o w a v e sp ecto m eter used. In gen eral, a k lystron w a s u sed
to g en erate m icr o w a v es. A n attenuator con trolled the input p o w er to the ca v ity and
44
w a s u su a lly set at 10 d B . A cry sta l d io d e w a s u sed to m easu re the ca v ity output,
i.e ., the p o w er reflected at the ca v ity freq u en cy (P c ). For c o n v e n ie n c e the entire
m o d e w a s d is p la y e d on the o s c illo s c o p e .
First, the z e r o -fie ld m ea su rem en t w a s
carried out. Pc w a s m onitored as a function o f tem perature in zero ap plied field . In
ev ery ca se the ca v ity w as kept und ercou pled (i.e ., im p ed ance o f ca v ity w ith sam p le
>
w ave
g u id e
im p e d a n c e ) at r o o m
te m p era tu re .
T h e tr a n sitio n
to th e
su p erco n d u ctin g state w a s m arked by a rapid drop in Pc a s the sa m p le w a s c o o le d
through the critical tem perature. T o carry out m easu rem en ts o f Pc as a fu n ction o f
the a p p lied dc fie ld , an au tom atic freq u en cy co n tro ller (M ic r o -N o w M o d el 2 10C)
w a s used to lock the klystron output freq u en cy to the ca v ity freq u en cy.
T h e signal
d irectly con trolled th e y -a x is o f an O m nigrahic 2 0 0 0 X -Y recorder.
F or lo w -f ie ld m e a su r em en ts ( / / 0H dc ^ 2 5 m T ) , th e d c fie ld w a s
gen era ted u sin g a pair o f h o m e-m a d e H elm h o ltz c o ils . Fig. 2 .3 s h o w s a sch em atic
o f the ex p erim en ta l set up u sed for the field d ep en d en ce m easu rem en ts. A K ep co
B ip o la r O perational P o w er S u p p ly /A m p lifier su p p lied the current to the c o ils and a
fu n ctio n gen erator (W a v etek M o d el 190) w as u sed to ramp the p o w er su p p ly. An
attenuator (H ew lett-P a ck a rd 3 5 0 ) w a s u sed to control the m ax im u m current to the
c o ils . T h e H elm h o ltz pair w as p o sitio n ed su ch that the sam p le, p la ced on the floor
o f the cop p er/b rass c a v ity , o c c u p ie d a region at the m id p oin t o f the tw o c o ils . T he
sa m p le s iz e w a s k ep t s u ffic ie n tly sm all su ch that the field in w h ich it w a s p la c ed
c o u ld be co n sid ered uniform . T he d c field w as oriented parallel to the length o f the
w a v e g u id e.
45
Matched Load
u
E
ok.
w
8.
i/i
>
s
ou
o
e
0>
o
u
03
E
u
J=
o
<N
(N
£P
£
CU CO
46
T3
X
2
3
or
E
-a
c
as
jy
13
L*
V
c
0/
c
£
s
o
•5
Cl
E
(0
tlO
T
3
TD
OJ
C /3
3
3
a)
3
C /5
c/3
On
a3
:/3
"c3
c
OJ
6
ka
t>
a
.
X
<u
a
j:
o
u
3
e
V
J=
o
c/i
ro
r-j
oh
LL
47
4
>
>
rt
S
_u3.
3
00
§
2
u
c
u
«
c
00
CO
E
u
-a
S
2
<u
-C
!n so m e c a se s j j ttH dt < 1.5T w a s n ecessa ry and w a s regu lated by a
W a lk er fie ld co n tro ller w h ich u sed a feed b ack from a rotating c o il gau ssm eter. In
this c a s e the d c field w as orien ted in the plane perpendicular to the a x is o f the w ave
g u id e . T h e error in th ese field m ea su rem en ts w a s ab out o n e p ercen t d u e to fin ite
field sw e e p in g tim e. In e v e ry ca se the sam p le w a s p la c ed on the flo o r o f the ca v ity
in the region o f m axim u m b rl, o f about 10 fi T .
T h e th in -film s and s a n d w ic h e d ty p e str u c tu r es u se d L aA IC >3
su bstrates. T h e film s [1 2 ] w ere ea ch p la ced on quartz p late w ith th e film su rface
fa cin g the quartz plate and m ou n ted against the cavity w all such that the quartz plate
fa c e d the c a v ity w a ll.
S in g le c r y sta ls w e re m o u n ted sim ila r ly . P e lle ts w ere
m o u n ted on sap ph ire rod, and the m easu rem en ts carried out by S. T y a g i at D r ex el
U n iv ersity on a com m ercial X -b an d sp ectrom eter [41].
2.3 C ryogenics
T h e c a v ity and part o f the w a v e g u id e w ere e n c lo s e d by a
sta in le ss steel ja c k e t and the en clo su re evacu ated (se e F ig. 2 .4 ). P ressure in sid e the
ca n c o u ld b e varied b etw e en a tm o sp h eric p ressu re and about 2 0 m Torr. T h e can
w a s im m ersed in the appropriate c o o la n t h eld in a d ew ar. L iq u id n itrogen (L N )
w a s u sed to c o o l to 7 7 K and by p u m p in g on L N tem peratures as lo w as 5 0 K w ere
a c h e iv e d . T o g et to 4 .2 K liq u id H e w a s u sed .
T h e rate o f c o o lin g c o u ld be
co n tro lled by con trollin g the lev el o f the co o la n t and the pressure in sid e the can . T o
m in im iz e tem p eratu re grad ien ts in the sa m p le, a co n tro lled am ou n t (ty p ic a lly 2 0 0
m T orr) o f H e g a s w a s let in sid e the ca n . S a m p le tem p eratu re w a s m o n ito re d
e m p lo y in g a calib rated co p p er-co n sta n ta n th erm om eter m o u n ted (u sin g m a sk in g
ta p e) in c o n ta c t w ith the o u ter sid e o f th e c o p p e r /b r a ss c a v ity in the reg io n
48
o cc u p ied b y th e sam p le. In the ca se o f m easu rem en ts on the 2 G H z sp ectrom eter a
carbon g la ss th erm om eter w as p laced in con tact with the alum inum ca v ity . Error in
tem perature v a lu es in both c a se s w a s estim ated to be about ± 2 K.
2.4 E x p e rim e n t
T o d eterm in e the T"" the sa m p le s w ere first
z e r o -fie ld -
c o o le d in earth's field (/J 0H < 3 0 //T ) . T he T d ep en d en ce o f m icr o w a v e absorption
w a s stu d ied by fo llo w in g the p o w e r r e flected at the c a v ity fr eq u e n c y ( P t ) as a
fu n ctio n o f tem perature. T h e z e r o -fie ld lo ss at any tem perature is then d efin ed by
P(0,T) = Pf (0 , T ) - Pt ( 0 , 7 , ) .
b e c o m e s in d ep en d en t o f T.
Tt m ark s the te m p era tu re b e lo w w h ic h Pt
In m o st c a s e s T L = 8 0 K ; h o w e v e r for a g g lo m er a tes
Tl = 6 5 K . F or 7 < 5 0 K , the lo s s d u e to the em pty ca v ity b e c o m e s com p arab le to
the lo s s d u e to the sa m p le. T h u s Pt ( 5 0 K ) rep resen ts the b ack grou n d absorption.
T he tem perature at w h ich a sharp drop in Pt. o cc u r s is the critical tem perature T™'
and m arks the o n set o f su p ercon d u ctivity. T o obtain 7’™"', from a typ ical zero -field
transition su ch as sh o w n in F ig. 2 .5 , o n e m ad e u se o f tw o line seg m en ts draw n as
sh o w n and id e n t ifie d th eir in te r s e c tio n p o in t w ith th e m ic r o w a v e c r itic a l
tem perature. T ab le III.3 lists the 7’"" for the d ifferen t sa m p les in this stu d y. T he
n orm alized p ow er absorption at any tem perature T is then
p
p ,( o ,n - p ,( o ,7 - ; )
pN
/>,(o,r;)-/,,(o,rI)
F rom th e z e r o - f ie ld
m e a su r e m e n ts an e s tim a te o f th e z e r o -
tem perature L on d on p en etration d ep th , A (0 ) , can be ob tain ed u sin g the tw o -flu id
m o d el and the m o d ifie d L on d on e x p r e ssio n for p ow er ab sorb ed b y a se m i-in fin te
slab o f su p ercon d u ctor in a m agn etic field . W e p ostp on e this d icu ssio n for Chapter
49
THERMOCOUPLE
tTO VACUUM PUMP
£ NEATER W I R E
FEEDTHROU&H
— LN* CAN VENT
LN* RESERVOIR
IR IS
ILH#
DEWAR
F ig. 2 .4 .
S ch em a tic o f the apparatus used to co o l the sam ple.
50
Ill o n ly sto p p in g to point out here that sin c e the m ax im u m ch a n g e in the p ow er
ab sorp tion o cc u r s w ith in a fe w k elvin o f the transition tem perature, the absorption
b e lo w 5 0 K is n eg lected w h ile m od elin g the transition.
For m a g n e to a b so r p tio n stu d ie s the sa m p le w a s first z e r o -fie ld c o o le d as d escrib ed a b ove to the desired tem perature T «
Tt . N e x t, to ob tain the
V C the d c field w as sw e p t m an u ally at rou gh ly 10'4 T /se c u pto
= 22 mT.
T h e re co rd ed d ata w a s the fie ld in d u c e d p o w e r a b so rp tio n o f th e sa m p le
P (H ,T ) - Pt ( 0 ,T ) .
T o ob tain the n o r m a liz e d fie ld -in d u c e d p o w er
absorption o n e w rote
AP ( H , T )
PN
P (\ H , T ) ~ P t ( 0 , T )
P, { 0 , T * ) - P( ( 0 ,7 7 )
H y s te r e s is in the m a g n e to a b so r p tio n w a s c le a r ly e v id e n t w h en
su bsequ en t to ob tain in g the V C the sam p le w as su bjected to a c y c lic m agn etic field.
T o record the h y ste r e sis lo o p s the m a g n etic fie ld w a s sw e p t lin ea rly b etw e en
^ 0H mas and - p 0H ma% (r e v e r se fie ld ), u sin g the s lo w e s t a v a ila b le sp e e d s , < 0 .9
m T /sec. Error in lo w -d c field m easurem en t w a s about ± 1 m T. M agn etoab sorp tion
in th e film s and the s in g le cy r sta ls w a s stu d ied w ith B rf in th e p la n e o f the
sp ecim en and B ( < 1 .5 T ),
51
(s^nm -qjB) °tj
from
AC susceptibility) at 91.5 K
S in c e m agn etoab sorp tion m easu rem en ts are u su ally m ade u sin g an
E SR sp ectrom eter in w h ich a m odulation field is u sed in addition to the dc m agn etic
fie ld , m a g n eto a b so rp tio n stu d ie s by a n um ber o f g ro u p s [4 4 , 4 5 , 4 6 ] h as o ften
b een co n d u c ted u sin g both field s. H o w ev e r, u n lik e spin reso n a n ce stu d ie s w h ere
the u se o f a m od u latin g field and a lock -in am p lifier for d etection in k n ow n to yield
the d eriv a tiv e o f the sig n a l, the c a se is not so sim p le here. F or e x a m p le , it w a s
sh o w n in R e fs 4 7 and 4 8 that in lo w m a g n etic fie ld s, w h ereas the direct ab sorption
sh o w e d no h y steresis, the use o f a m od u lation resulted in a h ysteretic sign al w h ich
w a s d e p en d e n t on the m o d u la tio n a m p litu d e!
In ou r in v e s tig a tio n w e h ave
o b serv ed [19] that the m icrow ave absorption in th ese sam p les is ex trem ely sen sitiv e
to as m uch as a fraction o f a m illi T e sla w h ich m ea n s that u sin g a m od u lation field
in ad d ition to the d c fie ld , h o w e v e r sm a ll, is b ou n d to in flu e n c e the sig n a l and
c o m p lic a te th e a n a ly s e s.
W ith th e se resu lts in v ie w , it b e c o m e s e s s e n tia l to
re c o g n iz e the pit fa lls o f u sin g the co n v en tio n a l m od u lation and lo ck -in d ete ctio n
tech n iq u e for . F ortun ately for u s, the sig n a ls are su ffic ie n tly stron g that o n e has
n o n e e d to resort to th is m eth o d .
T h e s ig n a ls re co rd ed h ere are th e d ir ect
absorption cu rves and no m od ulation field s are u se d .
53
Chapter III.
Results
In the past a sy stem a tic study o f the dc m a g n etiza tio n [4 9 , 5 0 , 3 9 ],
the lo w -f ie ld su sc e p tib ility [5 1 ] and th e z e r o -fie ld m ic r o w a v e r e sp o n se [5 2 ] o f
p o w d er sa m p les o f typ e-II m aterial has b een carried out in th is laboratory. It w a s
sh o w n that o n e c o u ld d escr ib e the lo w -fie ld su sc e p tib ility [5 0 ] and the z e r o -fie ld
(Z F ) m icr o w a v e lo ss w ith in the fram ew ork o f the L ond on [53] m o d el. T o accou n t
for th e Z F m icr o w a v e transition a sim p le m o d ific a tio n o f the m o d el w a s p ro p o sed
[5 2 ], T h e u se o f p o w d er sa m p les, as o p p o se d to sin tered sp e c im e n s a lso a llo w e d
fo r a sim p le co m p a r iso n b etw e en the resu lts o f the lo w -fie ld su s c e p tib ility and
m icro w a v e transition. Param eters, such as the L ond on penetration depth, extracted
from both data are in reason ab le agreem en t w ith ea c h oth er (s e e T a b le s. III. 1 and
III.2 ).
F u rth erm o re, the dc m a g n e tiz a tio n d ata p r o v id e d a str a ig h tfo rw a rd
con firm ation o f the M eissn er phase w ithin the grains.
W h ile all th ese data see m ed to fit w ell togeth er, the ob servation o f a
dc fie ld -in d u ce d m icr o w a v e lo ss [54] in p o w d ers ca m e as a surprise. In particular,
it co n tra d icted th e m a g n etiz a tio n data w h ich stro n g ly in d icated a M e issn e r state.
O n e n eed ed , th erefore, to u nderstand th is b reak d ow n o f su p e r c o n d u c tiv ity by a
lo w d c m a g n etic fie ld in th ese p o w d er H T S C sa m p les w h ich ap p eared , from all
other m easurem en ts, to be in the M eissn er state for the field regim e o f interest.
In th e f o llo w in g s o m e o f th e a b o v e r e s u lts o f the p r e v io u s
in v e s tig a tio n s are re c a p itu la te d .
F irst, the dc m a g n e tiz a tio n d ata and th eir
im p lic a tio n s are presented . N e x t, the lo w -fie ld su scep tib ility data and the fit to the
54
L on d on m o d el for n o n -in tera ctin g p articles is sh o w n .
F o llo w in g th is, the zero-
field m ic r o w a v e ab sorp tion study and the fit o f the data to th e m o d ifie d L on d on
picture is b riefly presented. T h e results o f the field -in d u ced m icr o w a v e absorption
study is introduced next. T h e top ic is d ivid ed into tw o sectio n s. In the first section
the o b se r v a tio n s on the virgin ab sorp tion is p resen ted .
T h is is fo llo w e d b y a
d is c u s s io n o f v a rio u s m o d e ls p r o p o se d , to a c c o u n t fo r th is lo w - f ie ld lo s s in
m ic r o w a v e p o w e r .
F in a lly , w e p resen t the RSJ m o d e l [1 6 ] and d is c u s s its
im p lica tio n s. T he o b serv a tio n s on h y steresis in the m icr o w a v e m agn etoab sorp tion
form the seco n d section and are the topic o f the n ext chapter.
3.1
DC M agnetization and Susceptibility
T w o o f the re su lts [4 9 , 5 0 , 5 1 ] o f p o w d e r sa m p le s on w h ic h w e
w ant to fo c u s are the fo llo w in g - 1) the z e r o -fie ld -c o o le d m a g n e tic iso th erm s (
co n sta n t T ) M ( H )
and 2) the tem p eratu re variation o f the in itial s u s c e p tib ility ,
= <?M
H-.0
A d etailed d escrip tion o f the apparatus and the Faraday m eth od used
to stu d y m a g n etiz a tio n can be fou n d in earlier reports from th is laboratory [3 9 ],
B r ie fly , the m eth o d in v o lv e d is as fo llo w s . T h e sa m p le in the form o f a k n o w n
am ou n t o f H T S C p o w d er in a # 5 g ela tin ca p su le . T h e sa m p le o f k n o w n m a ss,
ty p ic a lly a fe w hundred m illig ra m s, w a s then p la c ed in the cen ter o f tw o p airs o f
H e lm h o ltz c o ils o n e o f w h ich g en era ted a m a g n etic fie ld H z w h ile th e oth er
g en era ted a field gradient
az
1 . T h e c o ils w ere d e sig n e d as to y ie ld H = 0 and
d \i
—
= con stan t in the m id plan e. T he force F ,, on the m agn etic m om en t m o f the
<9z
sa m p le d u e to the fie ld w a s m easu red u sin g a C ahn 2 0 0 0 e le c tr o n ic b alan ce. T h e
m agnetization
M 7 o f the sam p le w as extracted from the force m easu rem en t u sin g
55
the relation F z = M —
1 w h ere a n om in al d en sity o f p = 6 .4 g /c m ^ fo r the
p dz
su p ercon d u ctor, ob tained from ideal unit ce ll com p u tation s, w a s u sed . H ere, m is
the m a ss o f the sa m p le. T he sa m p les w ere c o o le d through T c in zero (< 0.1 p T )
fie ld and data taken d u rin g th e su b seq u en t w arm up.
S a m p le tem p eratu re w a s
record ed u sin g a cop p er-con stan tan th erm ocou p le. Error in
w a s estim a ted to
be about 10% and that in tem perature to be ± 2 K.
F ig. 3.1 sh o w s the m a g n etiza tio n as a fu n ctio n o f fie ld for the 10
p m Y B C O p o w d er at 4 .2 K. It w a s fou n d that M ( H ) is e s s e n tia lly lin ear u pto
p 0H = 2 0 m T. T h e o b servation further ind icated that there is little or no h y ster esis
in the m a g n etiz a tio n in th is fie ld regim e. A t 7 7 K the lin ear r e g im e p rev a ils for
<10m T .
S im ila r ly , in th e c a s e o f th e 5 p m
B S C C O p o w d e r it w a s
o b serv e d that th e lo w -fie ld m agn etic isotherm at 7 7 K sh o w e d a linear d ep en d en ce
on field upto p 0H = 3 m T and n e g lig ib le h y ster esis (F ig . 3 .2 ). T h e o b se r v a tio n s
o f linearity and n on -h vsteretic b eh avior o f the m agn etic isoth erm s, ind icate that the
sa m p le, in th is fie ld region , e x ists in the M eissn er p h a se.
O n the other hand, in the ca se o f the 5 0 p m Y B C O a gglom erate, the
n o n lin earity as w e ll as h y ster esis w a s o b serv e d for fie ld s as lo w as 0 .8 m T at 4 .2
K (F ig . 3 .3 ) im p ly in g B?*0 in the sa m p le fo r q u ite m o d e st H v a lu e s.
In oth er
w ords, the o b serv e d d ev ia tio n from a linear b eh avior o f the M vs. H cu rve strongly
ind icates the p resen ce o f m agn etic flu x w ithin the bulk o f the sam ple.
56
'
50
■ i
.....
■
1
t
i
■
■r
■
E
40
□
□
□
i
30
■
i
n
EJ°
On
20
□
D°
10
ft
□B
|
1
1
1
.
1 -------- .
---- ■
* - ----- 1------- ‘■ ------------------------------>--------1
ftj ' / ---
0
10
20
30
40
i
50
Ho H (mT)
F ig . 3.1 Z F C isoth erm o f the 10 |im (grain d iam eter) Y B C O p o w d er at 4 .2 K.
N o te that M is linear to at least fi H ~ 2 0 m T su g g e stin g that B ~ 0 in the sam p le.
57
4
—
----1------
1—
------1------------- -------------1------------ ---------- 1
■
3
■
□
-
0
H
B
S
0
□
2
□
■
=?
'
■
\
0
l
□
□
_L_
..1
1
1
2
3
0
0
------ I
4
.
—
-------
1
5
6
Mo H (m T )
F ig . 3 .2 . Z F C m a g n etic isoth erm in 5 p m (grain d iam eter) B S C C O p o w d er at 7 7
K .O n ce again , n ote that M is linear in H up to p H ~ 3m T .
58
2.0
I 50 [xm YBCO agglomerate at 4.2K
-H0M (mT)
1.5
—
□
I
0
1.0 _
□
□
□
0. 5
1
-
0.0
s
□
0
1
2
HoH (mT)
3
F ig .3 .3 . Z F C iso th erm o f the 5 0 p m Y B C O a gglom erate sa m p le at 4 .2 K. N o tic e
that th e n o n -lin e a r ity in M
ca n be o b se r v e d at a s lo w a fie ld as 0 .8 m T .
59
It is u sefu l to bear th ese results in m ind w h en trying to understand
the m icr o w a v e respon se.
T h e s lo p e o f th e M ( H ) c u rv e in th e lin ea r r e g io n y ie ld s %m.
M easurem ents o f the initial su scep tib ility x in- *n several d ifferen t field v a lu e s, w a s
thus ob tained [51]. T h e o n set o f su p ercon d u ctivity ( T™*) is m arked by a n on -zero
X,n-
X,n s h o w s a grad u al in c rea se as T g o e s b e lo w
T™* and satu rates at
J a ck so n et al. o b se r v e d [5 1 ] that £ ,n(T ) in th e p o w d e r sp e c im e n s
s h o w e d a grain s iz e d ep e n d e n c e for T c lo s e to 7”™*.
T h is is u n d ersta n d a b le
b ecau se a rapid drop in A o cccu rs at the o n set o f su p erco n d u ctiv ity and the s iz e o f
the grain w as com p arab le to A . U sin g the so lu tio n to th e L o n d o n eq u a tio n s [53]
for a sp herical g eo m etry , a lo n g w ith the tw o -flu id tem perature d ep en d e n c e for the
L ondon penetration depth A (T ) (E q. (1 .2 1 )), the data w ere sh o w n [4 9 , 5 1 ] to be
d escrib ab le in term s o f the param eter R / A (0 ) , R b ein g the grain radius and A (0 )
the zero tem perature L on d on p en etration depth. Furtherm ore, S h a w and B hagat
s h o w e d that the o b serv ed j ,„ ( 4 .2 K ) versus the fit param eter R / A (0 ) fo llo w e d the
p red iction o f the L on d on m od el (se e F ig . 4 o f R ef. 4 9 ] p ro v id in g p r o o f o f internal
c o n s is te n c y .
—
F ig . 3 .4 s h o w s , a s an e x a m p le , the n o rm a lize d su sc e p tib ility data
as a fu n ctio n o f the n orm alized tem perature T /T " 13® in th e c a s e o f th e 10
x (0)
y
L
fdm Y B C O p ow d er sam p le w here the so lid lin e is a fit to th e equation
XiT)_
X(0)
[ l ~ ( 3 / * ) c o t h j + ( 3 / * 2)]
[ l - ( 3 / * 0)cothjr0 + ( 3 / x o j ]
w ith x = [ /f /A ( T ) ] and x 0 = [/? /A (0 )].
T h e data sh o w s g o o d agreem en t w ith the
m od el ex cep t for tem peratures c lo s e to T"13® w h ere Jackson et al. d em on strated that
60
c o n s id e r in g
a d istrib u tio n in T™p [5 1 ] p ro d u ced a b etter fit. S in c e T"1^
d eterm in ed em p ir ic a lly , Eq. (3 .1 ) has o n ly o n e param eter
jc0 ,
is
u sin g w h ic h A (0 )
w a s d eterm in ed .
S h a w et al. a ls o ob ta in ed a sim ila r fit fo r the initial su sc e p tib ility
data o f B i4 C a 3 S r 3 C u 4 0 1 6 p o w d er su percon d uctor [4 9 ]. T a b le III. I lists o f som e
o f th e s a m p le s, th eir r e s p e c tiv e grain s iz e s , the fittin g p aram eter jr„ and the
calculated A (0 ) valu es from the m agn etization data.
T h e v a lu e s fo r the p en etration d ep th s for Y B C O a lo n g the c -a x is
and the a and b a x e s are k n o w n to b e A( (0 ) = 8 0 0 n m and Aa ^(0 ) = 160 n m
r e sp e c tiv e ly . In co m p a r iso n , the num bers listed in T ab le III. 1 appear to b e spread
about in th is range. T h is is to be e x p e c te d sin c e in p o w d ers o n e has a d istribu tion
in grain o rien ta tio n s lea d in g to an a v era g e v a lu e for A (0 ).
V a r io u s k in d s o f
avera g in g [55, 5 6 ] su g g e st that the range o f A (0 ) can be an yw h ere b etw een A wA(0 )
to A ( 0 ) = 0 .8 A( ( 0 ) , that is ro u g h ly b etw e en 0 .1 6 p m to 0 .6 p m . O n e n o tes that the
v a lu es extracted for A (0 ) from the m a g n etiza tio n data fa lls c lo s e to this range. A
g o o d a g reem en t w ith this m od el w a s a lso obtained for the c -a x is align ed p ow der.
61
Table
I U .l.
List of samples used in the m agnetization and
susceptibility study [49, 51] along with their respective grain sizes,
the parameter x0 = R/A(0) used to fit their susceptibility and the
extracted values for the zero-temperature penetration depth Am‘g(0).
Sample
average
x 0 = R / A (0 )
A""'*(0)*
grain
(fit
( n m)
dia.
parameter)
2 R (|im )
Y B C O p ow d er
1
4
130
Y B C O p ow d er
2
4 -5
220
Y B C O p ow d er
5
6 -7
380
Y B C O p ow d er
10
7 -8
700
BSC C O
5
6
500
p o w d er
Y B C O c-a x is
2
* 0 = /f/A u, ( 0 ) = 1 2
Auft(0) ~
x 0 = /?/A ,.(0) = 5
A( ( 0 ) - 4 0 0
160
a lig n ed pw dr
* Error in A"“ K(0 ) ~ ± 20% o f tabulated value.
62
0.6
10 p m YBCO powder
O
London Model
06
C
X
04
O-l
0 2
0 3
0 4
0 .5
0 6
0 ?
OB
0 3
T
F ig . 3 .4 . N o r m a liz ed initial su scep tib ility as a fu n ction o f the red u ced tem perature
for the 10 p m Y B C O p o w d er in ap p lied field o f 1 m T [3 9 ], T h e so lid lin e is
o b ta in ed from the L ond on m od el (E q .(3 .1 ) and the tw o -flu id m o d el fo r AfT).
63
3.2
Zero-Field M icrowave A bsorption
It is w ell k n ow n [5 3 ] that at the o n set o f su p e rco n d u c tiv ity a sharp
d ro p in th e m ic r o w a v e p o w e r a b so r p tio n o f th e s a m p le o c c u r s .
T h is is
u n d erstan d ab le sin c e the L on d on penetration depth d e c r e a se s rapidly as T drops
b elo w T c and as a result th e electro m a g n etic field s w ith in the sam p le are e x p e lle d .
T h e critical tem perature T ”K is d eterm in ed by d ra w in g tw o lin e se g m e n ts to the
transition cu rv e as sh ow n (se e F ig. 3 .8 , for e x a m p le).
It is w e ll re c o g n iz e d that the sh ape o f the transition cu rv e d ep en d s
on th e q u a lity o f the sa m p le , g e n e r a lly b ein g b road er for the m o re gran u lar
s p e c im e n s. A . G o u ld e t al, [5 2 ] co n d u c ted a sy stem a tic in v e stig a tio n o f th e zerofie ld m ic r o w a v e ab sorp tion o f p o w d er sa m p les o f typ e-II m aterial as a fu n ction o f
the grain s iz e . T h e ex p er im en ta l set up and m eth o d d escr ib ed in C h ap ter 2 w a s
f o llo w e d .
T h e y d is c o v e r e d that th e tem p eratu re d e p e n d e n c e o f the p o w e r
absorption w a s co n tro lled by the s iz e o f the su p ercon d u ctin g grain. T h e grain siz e
in their study w as varied b etw e en 2 p m and 10 p m and it w as fou n d that the larger
the grain radius the narrower w a s the transition. T h ey a lso found that the ch a n g e in
ab sorp tion for T < 0 .8 T c w a s n e g lig ib le . T h e study a lso in clu d ed a p o w d er sa m p le
m ad e o f a g g lo m e r a te s 5 0 p m in d ia m ete r w h ic h s h o w e d a s ig n ific a n tly w id er
tr a n sitio n , c o m p a r a b le to that o f m a n y sin tere d s a m p le s .
T h e b r o a d e n in g ,
p r e su m a b ly , w a s c a u se d by the p r e se n c e o f the m an y intragran u lar in te r fa c e s
w ith in the grain s.
sp ecia l
In an oth er sa m p le w h ich w a s c o m p o s e d o f 1 0 0 p m g ra in s,
care w as taken to a v o id agglom eration and a m u ch sharper transition w a s
ob tained. H o w ev e r the th eoritically p redicted instantaneous drop in absorption w a s
not realized , perhaps, b ecau se it w a s not p o ssib le to en tirely elim in a te intergranular
interfaces.
64
A s ig n ific a n t result o f the investigation reported in R ef. 52 w as to
s h o w that the sh a p e o f the tran sition cu rv e in th e c a se o f the n o n -a g g lo m e r a te
g ra in s is w e ll d e s c r ib e d by
a sim p le e x te n s io n o f L o n d o n 's [5 3 ] th eo ry
as
fo llo w s . T h e su rface im p ed ance in term s o f the propogation con stan t /? is
(3 .2 )
w here
( 3 .3 )
From E q. (3 .3 ), w e see that in the n orm al-state w h ere A is in fin ite, the ab sorp tion
is d eterm in ed by 8 .
F or p N = l / r £ 2 - m , 5 N(1 0 G H z ) = 5 /r m . H o w e v e r , in the
c a se o f grain s o f radius R w ith a lo w norm al state c o n d u c tiv ity , o n e can h ave 8
m u ch larger than R fo r all T.
In such a c a se , it can be argu ed that the rela tiv e
m icr o w a v e p o w er absorbed P j P N in the norm al state is co n tro lled b y R rather than
8 , the elec tro m a g n e tic skin depth. A n d sin c e A (0 ) s 0 .2 // m , b e in g m u ch sm aller
than th e rad iu s R s o that, it co n tr o ls th e ab sorp tion in the su p e rco n d u c tin g state.
S o o n e can rew rite the L on d on e x p r e ssio n for the relative p o w er ab sorp tion o f a
se m i-in fin ite sla b (E q.( 1.53) for th e c a s e o f sp h eres, in term s o f the radius R and
A as
( 3 .4 )
65
T h e data w ere sh o w n to be in e x c e lle n t a greem en t w ith the m o d ifie d L o n d o n
e x p r e s s io n and is re p r o d u ce d here in F ig. 3 .5 .
T™w b ein g e x p e r im e n ta lly
d eterm in ed , the zero-tem p erature value for A, A (0 ), w a s extracted u sin g the tw o flu id tem p eratu re d e p e n d e n c e for the penetration depth (E q. (1 .2 1 )). T h e s e h ave
b een liste d in T a b le 3 .2 for the so m e o f the sa m p le s stu d ied .
C o m p a rin g the
n u m b ers w ith th e co r resp o n d in g o n es listed in T a b le III. 1 w e n ote that th ere is
g o o d a g r e e m e n t o f b o th
A (0 ) v a lu e s a s w e ll a s th e p a r a m e te r
R j X (0 ).
In cid en tally, it sh ou ld be noted here that if instead o f the tw o flu id m od el on e c h o se
the M iih lsc h leg e l [5 7 ] form w h ich is g iv en by
( 3 .5 )
fo r the tem perature reg io n c lo s e to T c , the o n ly c h a n g e w o u ld b e th e red u ction o f
the e ffe c tiv e A (0 )b y V 2 . S in ce the tw o -flu id picture appears to y ie ld a co n sisten t
d escrip tio n o f th ese p o w d ers, for the rest o f the a n a ly sis the tw o -flu id tem perature
d ep en d en ce for penetration depth has been m aintained.
T h e m icr o w a v e transition , as can be seen from th e fits (F ig . 3 .5 ), is
rea so n a b ly w e ll rep resen ted b y th is m o d ific a tio n .
T h is is so m e w h a t su rp risin g
b eca u se, in reality, the grain s are not all sp heres and the grain s iz e d istrib u tion is
not in sig n ifica n t (se e F ig. 2 .3 ).
66
Table III.2.
HTSC
Zero-Kelvin London penetration depth of some powder
sam p les,
obtained
transitions [52] to Eq. (3.4).
from
fitting
zero-field
The values compare well with that
obtained from susceptibility data (see Table III.l).
Sample
average
jc0 = R / k ( 0 )
grain
dia. (pm )
Amtt(0 )
(A )
(fit
paramete
r)
YBCO
2
3 .5
2800
3
4
3500
6
6
5300
10
8 .5
6000
YBCO
2 x 2 x
5
2000
aligned
1.5
p o w d er
YBCO
p o w d er
YBCO
p o w d er
YBCO
m icrow ave
p o w d er
p o w d er
67
1.0
0.8
0.6
0 .4
0.2
2 .5
o-
r— U y E T ^ S T E ^ t a - j B — B - 3
0 .9 0
0 .8 5
fTTD°
0 .9 5
6
1 .0 0
T /T c
F ig . 3 .5 . Z ero fie ld m ic r o w a v e tran sition s o f m icron s iz e d Y B C O grain s at 10
G H z from R ef. [5 2 ]. T h e fit w a s o b ta in ed u sin g the m o d ifie d L on d on m o d el and
the tw o -flu id tem perature d ep en d e n c e for the penetration depth. T he fit param eter
/? /V 2 A (0 ) is in d icated for the variou s p ow d er sa m p les (i) 2 p m Y B C O (O ) (ii) 3
pm
YBCO
(x )
(iii) 6 p m
YBCO
( A ) an d (iv )
68
10 p m
YBCO
( ).
An alternative approach to m o d ellin g the zero-field m icr o w a v e transition
o f th ese p o w d ers has been su g g ested by Ferrell [58] w h o co n sid ered the sam p le to
be c o m p o s e d o f su p e r c o n d u c tin g sp h e r e s. T h e m o d e l is an e x t e n s io n o f the
L o n d o n so lu tio n fo r a sp h ere in a sta tic fie ld , w ith the e x te n s io n to n o n -z e r o
freq u e n c ie s b ein g m ad e by a ssu m in g X is a c o m p le x q uan tity.
It turns out that,
although in p rinciple the Ferrell m odel sh ou ld g iv e a better fit sin c e it has ex p licitly
taken into acco u n t the shape o f the grain s, the data appear to agree better w ith the
sim p le interp retation o f R ef. 5 2 (p resen ted a b o v e ). A fit o f the data to Ferrell's
m o d el is sh o w n for so m e o f the p o w d er sa m p les in F ig s. 3 .6 a, b and c.
69
Ud/d
70
CD
CO
o
O
CO
O
CM
U d /d
71
t“
O O
o
& P
=t
o
u
E
X
V
2
’u
N
<u
JO
■o>
ro
M
II
4>
«
tj
>
S>
«
c
o
a.
M
a
3
'n "
X
O
o
u
•o
2
Cl
o
o
u
m
>
u.
a
43
^5
E
2
power
sphere, with fit parameter R / \ { 0) = 9 {— ), 10 (
for
■0
m od el,
JO
[58]
1/1
c
Ferrel
) and
by
a
1 1 (— )
absorbed
+
>>
superconducting
o
c
o
the
o
so
U
o1/;
«
J=
E—
using
E
u
c
obtained
(—
fits
s.
are
LO
co
CO
o
Ud/d
72
E
3.
o
o
c
o
w
0>
>
*
O
uU
E
£ P
=t.
0
rn
1
o
=J.
V
U
N
fib
tl
<U
r3
O
s.
O'
y.
O
t/s
.O
■o
<u
4—1
■5
c
c
o
O.
c3
■o
X
N
o
o
u
■o
o
o
CL
u
U
o
sO
ro
>*
u
jU
o3
E
CQ
■5
2
sphere, with fit parameter
R ( \ ( 0 ) = 16 (— ), 18 {
) and 20
using the Ferrel [58] model, for power absorbed by
u*
a superconducting
C
’3u .
OJO
The solid lines are fits obtained
in
o
From the stu d ies on p ow d er sa m p les d isc u sse d a b o v e w e see that a
c o n siste n t d escrip tion o f th eir m a g n etiza tio n and m icr o w a v e p o w er absorption is
ob tain ed w ith in the fram ew ork o f the L ondon m od el. In other w ords both the Z FC
m a g n e tiz a tio n and the m ic r o w a v e ab sorp tion stu d ie s in d ica te that th e p o w d ers
b eh a v e as local L on d on su percon d uctors and for the field region o f current interest
the sa m p les appear to in d icate near p erfect field screen in g . T a b le s III. 1 and III.2
s h o w s the v a lu e s o f fit param eter R / k { 0 ) ob ta in ed from the tw o m eth o d s are in
g o o d a g reem en t as are the corresp on d in g penetration dep th s. H o w ev e r, as b efore,
it has to be p o in ted ou t that the v a lu e s for the p en etration d ep th s ob ta in ed from
th ese m easu rem en ts on p ow d er sa m p les corresp on d on ly to an average value.
T h e z e r o -fie ld resp on se o f the p ow d ers have b een ob ta in ed several
tim es o v e r the years (1 9 8 8 -1 9 9 5 ) and h ave sh o w n little or no c h a n g e in m o st ca se s
im p ly in g th e p o w d e r s are q u ite sta b le. F ig . 3 .7 b is th e r e c e n tly o b se r v e d
m ic r o w a v e p o w er ab sorp tion for the 2 p.m Y B C O p o w d er at 10 G H z. T he so lid
lin e is a fit to Eq. (3 .4 ) y ie ld in g A (0 ) = 2 3 3 nm . F igu re 3 .8
in the c a se o f the 5 0 Jim Y B C O a g g lo m era te at
results fo r the 5 |Lim B S C C O p o w d er at
for th ese sa m p le s
10 G H z.
s h o w s the transition
10 G H z and F ig. 3 .9 s h o w s the
T h e critical tem perature 7’"!K
is listed in T a b le III.3 a lo n g w ith the Ami' (0 ) e stim a te s.
T he
v a lu e s sh o w g o o d a greem en t w ith that ob tain ed from the su sc e p tib ility data. O ne
can a lso see that the 7’“” appears to d ecrea se so m e w h a t w ith in creasin g freq uency.
S u ch a d ecr ea se in T**' w ith in crea sin g 0) has a lso b een o b se r v e d in the ca se a
Y B C O thin film on L a A 1 0 3 , s in g le crystal o f E rB C O , a tw in n ed Y B C O sin g le
crystal as w e ll as in the san d w ich P rB C O /Y B C O /P rB C O (s e e T ab le 2 in R e f [17]).
In a ll c a s e s the tra n sitio n w a s o b se r v e d to be fairly narrow .
T h e m ic r o w a v e
transition in the ca se o f P rB C O /Y B C O /P rB C O at 10 G H z is sh ow n in F ig. 2.5 . In
73
the c a s e o f the 10 p m Y B C O p o w d e r th e m ic r o w a v e tra n sitio n h as g ro w n
s o m e w h a t w id e r and a ls o d e v e lo p e d a n o tic e a b le p eak at th e o n s e t o f the
su p erco n d u ctin g transition. F igure 3 .1 0 sh o w s the m ic r o w a v e transition in the 10
p m Y B C O p o w d er at 10 G H z. T h is p eak in the ab sorp tion at T™*' has a lso b een
o b serv ed to appear o v e r tim e in a fe w other c a se s as w e ll and has b een sh o w n [59]
to be c o n tr o lle d by the sa m p les p a ck in g d en sity . T h e T ”w, h o w e v e r , rem ain ed
u nchan ged to w ithin the error in tem perature m easurem ent.
74
75
O
<U
XL
(-
O
N
X
0
o
h
£
C-l
1
l.
V
o
E
m
■5
c
c5
t_
o
3
cd
«
wj
fc?
H
O
S
<
c
o
CA
c
cd
O
<D
<U
(s:
<
O
r^i
m
fNl
II
©
v.
>
2
S
O
ka
O
•o
c
cd
OJ
cd
E
2
rn
o
ua
u
c
t
O"
s
X;.
76
r -
Cd
y:
e»4
Uu
C
QJ
77
in the micron sized p ow ders.
78
C
•a
L.
<D
■o
o
Q.
o
u
m
>"
E
a.
©
T
3
C
cd
E
3
IS
i_
o
c
o
1 /3
c
PJ
a,
E
3
c
o
C/3
c
cd
-C
00
m
<N
(sjiun qjB) ^
o
oW*
o
! /3
©
r3
s
3
rn
f2»
(X
79
8
c
o
E
2
aj
o
Q
J
C/3
_o
cj
C/3
U
3
CJ
<u
>
CL
E
cd
c/3
<U
oj
<u
H
•3
IS
8.
-3
X)
cd
a>
CJ
o
c
[59] to be controlled by the packing density of the grains in
©
J=
shown
N
X
a
u
i*
c/3
Cd
Table 111.3.
List of some of the samples used in the present study
along with their respective microwave critical temperature T™", and
the London penetration depths obtained from zero-field microwave
transition.
Sample
average
measurement
grain dia.
Frequency
r B(0 )
(K)#
n in
10
8 9 .7
860
22
8 9 .5
760
36
8 5 .3
740
60
8 8 .3
730
2
10
8 6 .5
233
5
10
105
466
(pm )
YBCO
y ffll
10
a)fin
(GHz)
p o w d er
YBCO
p o w d er
(A K Z O )
BSC C O
p o w d er
# Error in 7 7 “' - ± 2K .
80
3.3
Field Induced Microwave Absorption - Virgin Curve
T o study the lo w -fie ld induced m icr o w a v e absorption (L F IM A ) the
sa m p le, as d escr ib ed in C hapter 2, is first ze ro -fie ld c o o le d (Z F C ) to the d esired
tem p era tu re T « T C (th e m ic r o w a v e tran sition tem p eratu re; th e su p e rscr ip t is
a v o id e d fro m here on ) and th en su b jected to a s lo w ly ram ped d c m a g n e tic field .
T h e m a xim u m dc field am plitude w a s / i 0H max = 2 5 m T . T h e resu lt is an in crea se
in th e p o w e r ab sorp tion b y the sa m p le w h ich has b een the su b ject o f n u m ero u s
stu d ies [6 0 - 6 3 ] and in a variety o f sam p les.
T h is lo w -f ie ld in d u ced m ic r o w a v e a b sop rtion (L F I M A ), as has
been m en tion ed b efore, can be broadly cla ssified under tw o d istin ct ca teg o ries - (1 )
th e v irg in a b so rp tio n c u rv e or virg in cu rv e (V C ), i.e ., th e a b so rp tio n on first
e x p o su r e to the dc fie ld and (2 ) th e h y steretic ab sorp tion that o c c u r s w h en the
sa m p le is c y c le d through the dc fie ld su b seq u en t to o b ta in in g the V C . T h e m o st
im portant d istin ction is that the V C is m easured on first ex p o su re to the field.
In th e f o llo w in g , the resu lts o f (1 ) are p re se n te d .
W e h ave
co n ce n tra te d on Y B C O p o w d ers o f grain d iam eters 2 p m , 6 p m , 10 p m , Y B C O
a g g lo m e r a te s m ad e o f p article o f 5 0 p m and 88 p m in d iam eter, B S C C O p o w d er
o f 5 p m grain s and an a lig n e d 2 p m Y B C O sam p le. In ad dition L F IM A o f Y B C O
p e lle ts [6 4 ] h a v e a lso been m easured . S a m p le preparation and m ou n tin g h as b een
d escrib ed in C hapter 2. T ab le II. 1 lists the various sa m p les in clu d ed in th is study.
T h e V C , rep resen ted by the sc h e m a tic in F ig . 3 .1 1 , is the initial
resp o n se to an ap p lied field . P resum ably, here the sam p le has no m agn etic h istory.
T h e tw o strik in g featu res o f the V C , are i) the l i n e a r rise o f a b so rp tio n at lo w
81
a p p lied d c fie ld s and ii) the saturation in ab sorp tion as field is in crea sed . T h e ya x is o f the V C is the field ind u ced p ow er absorbed AP ( H , T ) = Pt ( H , T ) ~ P t ( 0 , 7 )
and the x -a x is m arks the externally applied dc m agn etic field ^ „H w h o se m axim u m
value is d enoted
F ig. 3 .1 2 is the V C in the c a se o f the 10 |im Y B C O p o w d e r at 5 0
K , 10 G H z. F ig. 3 .1 3 is the V C o f th e sa m e 10 p m Y B C O p o w d e r at a lo w e r
tem perature o f 4 .2 K and 10 G H z. T h e V C o f the 10 p m Y B C O p o w d er at the
lo w e s t m icr o w a v e freq u en cy o f 2 .5 G H z and 7 7 K is sh o w n in F ig. 3 .1 6 .
3 .1 8 is that o f a Y B C O p e lle t at 10 G H z and 7 7 K.
VC o f 50 pm
F igu re
YBCO
a g g lo m erate at 3 6 G H z and 7 7 K is sh o w n in F ig. 3 .1 9 . For co m a p r iso n , the e ffe c t
o f the field on an Y B C O thin film [1 2 ], P rB C O /Y B C O /P rB C O san d w ich as w e ll as
s in g le cr y sta ls o f Y B C O and E rB C O [1 7 ] h ave b een stu d ied . It w a s o b serv e d that
m agn etoab sorp tion in c -a x is thin film s and s in g le crystals at T a fe w k elv in b e lo w
T c s h o w v ery little s e n s itiv ity to fie ld s le ss than 5 0 m T .
T h e v a stly d iffer en t
ch a ra cteristics o f the dc field in d u ced lo s s in thin film s as co m p a red to p o w d ers
[1 7 ] and sin tered m aterials [6 5 ] can be seen in F ig. 1.2 on p a g e 5. In th e c a s e o f
thin film s the lo s s is not o n ly linear up to 1.5 T but a lso n o n -h y ste retic and, as
s h o w n in R e f. 6 6 th is fie ld -in d u c e d ab sorp tion is w e ll d e scr ib ed by c o u p lin g o f
m icr o w a v e currents to flu x o n s.
82
X
eg
c
c /i
w
13
cE—1 e*
ea
S j=c«
'—'
"O
<u
is
o
<U
-c
a>
.a
o
•O
Z
c
o
4 -J
e*
oC/J
X
eg
u
>
u *o
x
«g
s
"O
<
u
c/2
eg
K
CJ
c
2
u
x
■o
.Si
"S.
Qeg
e/3
eg
c=
o
eg
b.
3
13
c /i
■o
G
eg
eg
E
2
U
<t>
X
o
C/)
<U
PNJ
O
C^nm qje) CHJdV
83
cn
W)
LE
4>
o
u
of the dc field sw eep .
eg
OJ
o
c
y
c
the maximum
e*
oc/i
X
eg
Y B CO powder (grain diam eter-1 O^m) at 10GHz, 50K
to
-4—»
c
3
-Q
0
2
4
6
8
10
12
14
16
18
p0H (mT) — ►
F ig . 3 .1 2 .
F ie ld d e p e n d e n c e o f th e virgin m a g n eto a b so rp tio n for the 10 (im
(grain d ia m e te r ) Y B C O p o w d er at 10 G H z
and 5 0 K.
T h e s o lid lin e is the
represen ts the fit ob tain ed u sin g Eq. (3 .6 ) w ith t l 0H 0 = 2 .2 m T .
84
85
86
with
(sjiun qjE) (H)dV
jiq
H q =0.5 m T.
□ □
IA
Uj
c
ID
<N
u
sz
hr-~
r~~
M
■o
c
r t
N
X
in
a
(N
H
£
CM
<3
—
Cj*
E
rS
C/l
4>
«
4 -^
£>
E
_o
W)
60
CJ
m
O
u
co
>•
E
o
v>
<—
O
H
E
r*">
wi
It
afc
=1.
U
>
i
iri
( s jiu n q iB ) ( H ) d V
cn
00
Ll
87
S'
CT"
LU
(sjrnn qre) ( h )JV
88
(sjiun 'qjB) (H)dV
Eq. (3.6) (sec Table III. 4 for fit param eters).
00
oc
n
-a
c
n
N
I
a
o
rt
<N
CM
CO
JZ
sco
=3.
©
co
6
Z
0)
"a,
E
S3
line and was
ft
—
"2
“
c.
1/5
o
U
CQ
>•
ri
5<~.
-C
c/1
00
■
o
©
o
-
U
>
= i)
(s^ran
£
(H)av
90
obtained using Eq. (3.6). (See Table 111.2 for
rj
-C
H
"3
55
a
A
"3
^
-y
-,/;
E
ca
*
&.
C.
2
/■,
3
rd
T3
'■”
■A
O
-j
—
-3
>
91
the fit obtained
using Eq. (3.6) with
H 0 = 4.5 m T.
92
A. G ou ld el al. w ere the first to sh o w [67] that the V C co u ld be fit
to a sim p le one-param eter em pirical equation o f the form
A P (H )
PN
A P (H )
a
H /H „
( 3 .6 )
1 + H /H
m easures the ch an g e in absorption on ap plication o f a dc field H , a
sets the s c a le and m ea su res the h igh fie ld or th e satu ration p o w e r a b so rb ed is
rep resen ted as a fraction o f the norm al state lo s s w h ile H 0 is a fie ld param eter
represen tin g the field required for o n e -h a lf the saturation p o w er ab sorb ed . B oth a
and H 0 d ep en d on the m icro w a v e freq uency <w/27T and the sam p le tem perature T.
For all the sam p les it w as found that the L FIM A V C w a s extrem ely w ell d escrib ed
by the em p irical form Eq. (3 .6 ). In F igs. 3 .1 2 to 3 .2 0 , the so lid lin e is a fit to the
a b o v e eq u a tio n . T ab le III.4 lists the fit param eters for E q. (3 .6 ) for the variou s
sa m p le s, a is g iv e n as a p ercen t o f the ju m p ( Pt ( T [. ) - P( { 7 7 )) in the z e r o -fie ld
a b so rp tion as the su p e rco n d u c tin g tran sition is traversed ( c f F ig . 2 .5 ).
T y p ic a l
v a lu es o f a is about 10% o f P N in the ca se o f the p ow d ers at 7 7 K and as m uch as
60% in the ca se o f the agglom erates.
93
Table IU .4.
S am p le
Sample characteristics and parameters for Eq. (3.6)
T ype
A v . grain
(D/lTt
T (K )
siz e ()im )
N o.
1
(G H z)
YBCO
2
*
(m T )
a
(% ) +
(K )#
4 .2
4
4
30
4
7
77
2 .2
8
2 2 .5
77
1.8
7
36
4 .2
3 .6
4
77
1.5
2
8 6 .5
10
p w d r.
(A K Z O )
2
YBCO
6
10
88
77
1 .2
7
10
2 .5
92
77
3 .3
54
10
89
4 .2
4
9
50
2 .2
10
77
0 .8
20
81
0 .6
22
84
0 .5 2
21
86
0 .5 6
10
88
0 .5 5
3
p w d r.
3
YBCO
p w d r.
94
Table. III.4. VC param eters (contd).
S am p le
T ype
A v. grain
( 0 / 2 Jt
T (K )
(m T )
s iz e (Jim)
N o.
3
YBCO
10
^0^0 *
a
(% ) +
(G H z)
(K )#
2 2 .5
8 9 .5
77
0 .8
8
36
87
4 .2
1 .9
2
50
1.2
2
77
0 .4
3
p w d r.
8 8 .3
60
4
YBCO
50
2 .5
77
0 .0 1
77
6 .5
64
77
5.1
56
2 2 .5
77
5 .3
4 3 .5
36
77
4 .5
25
aggm
10
5
YBCO
92
88
10
87
77
1 0 .4
2
36
90
77
67
10
9 0 .5
8 .8
3
10
9 0 .5
8 .8
2 .4
19
2 .3
2 8 .3
1.9
70
0 .5
aggm
6
YBCO
aligned
pwdr
7
YBCO
p ellet 1
8
YBCO
p ellet 2
95
Table. III.4. V C parameters (contd).
S am p le
T ype
size (p m )
N o.
9
A v. grain
BSC C O
5
yfttVs
T
00^0
a
(K )
(m T )
*
(% )
t
4 .2
1.5
15
50
0 .7
16
77
0 .4
15
2 2 .5
77
0 .5
13
36
77
(0/2 k
(G H z)
(K )#
10
105
pw d r
# Error in T em perature ± 2 K.
* Error in p 0H 0 = ± 0 . 2 m T .
t Error in a - 10% o f the tabulated value.
96
3
A n im portant d isc o v e r y [17] w a s m ade w h en it w as fou n d that the
n o rm a lize d m a g n eto a b so rp tio n AP/ a
v e r su s the fie ld ratio
H /H () y ie ld e d a
u n iv ersal cu rve for all granular sa m p les irresp ective o f the grain siz e , freq u en cy,
tem perature or exten t o f n on-linearity in the Z FC m agn etic isotherm . T h is universal
v ir g in
c u r v e is sh o w n in F ig . 3.21
and c le a r ly d e m o n str a te s th e g en er a l
ap p lica b ility o f Eq. (3 .6 ) (as an ex a m p le, data o f a Y B C O p ellet sam p le o f G iura et
al [6 0 ] at 4 8 G H z and 1.5 K has a lso been fit to Eq. (3 .6 )). In all c a se s w e see that
the V C rises sharply at lo w fie ld s and sh o w s saturation at m od erate field valu e (=
2 0 H 0 ).
From the v a lu e s o f the fit p aram eters H 0 and a listed in
T ab le III.4, the ob servation s can be ca tegorized as fo llo w s —
( 1) there is no system atic correlation b etw een H 0 and the nom inal grain size
(2 ) for fix ed (Q and increasing temperature:
(i) H 0 d ecreases m on oton ically and
(ii) a is rou gh ly con stan t fo r T < T c . A s T ap proaches T c a sh o w s
a rapid d ecrease.
F ig . 3 .2 2 s h o w s the o b s e r v e d tem p era tu re d e p e n d e n c e o f th e fie ld
param eter H () p lotted a s fu n ctio n o f the tem perature T for sev era l m ic r o n -siz e d
p o w d e r
a
a s
T h e
a
a n d
a g g l o m e r a t e
f u n c t i o n
o f
v a r i a t i o n
s a m p l e s .
t e m p e r a t u r e
i n
t h e
in
Fig.
t h e
c a s e
c a s e
Y o . 9 C a o . l S r B a C u 2 . 9 5 C o o . 0 5 0 7 - y
3 . 2 3
o f
o f
a t
1 0
s h o w s
t h e
a
GHz
1 0
97
) t m
v a r i a t i o n
YBCO
s i n t e r e d
is
(3 ) for fix ed tem perature and increasing (Q .
(i) H„ d ecreases slig h tly but
t h e
s h o w n
o f
t h e
p o w d e r
s c a l e
a t
1 0
p a r a l l e l o p i p e d
in
Fig.
3 . 2 4 .
f a c t o r
G H z.
o f
(ii) a ex h ib its a dram atic drop, the signal b eco m in g alm ost
u n o b ser v a b le at 6 0 G H z,
F ig. 3 .2 5 a e x h ib its th e o b se r v e d variation o f the s c a le fa cto r «
fu n ctio n o f the freq u en cy
w e o b serv e
co/
2
k
as a
for the ca se o f the 10 p m Y B C O p o w d er w h ere
a to d ecr ea se from rou gh ly 55% at 2 G H z to <0.1 % at 6 0 G H z.
(4 )
b oth
a
and
H 0 are m u ch larger in th e a g g lo m e r a te s , p resu m a b ly
reflectin g the p resen ce o f additional boundaries.
1.0
0.9
1
1
b
a
h
§
f
T
B
C
D
E
F
G
H
1
J
a
—
0.4
T
>
g
5
"
i
0.1 L
I
<
123 pellet @ 8.8 K
123 p e lle t® 19.1 K
123 pellet @ 69.8 K
123 p e lle t® 28.3 K
YBCO pwdr (2 pm)
YBCO pwdr (6 pm)
YBCO agg (50 pm)
YBCO agg (88 pm)
BSCCO pwdr (5 pm)
YBCO pellet (Ref. 3)
YBCO aligned pwdr (2 pm)
A
0.5
0.2
o i
r
J*
_
0.7
0.3
b f e e abc'md
■
-
0.8 —
0.6
I
n
t
%
5
l
10
1----------_
15
1
20
1... ---------- 1--------------- I
25
30
35
H/ H 0
F ig . 3 .2 1 . T h e u n iv er sa l a p p lic a b ility o f E q. (3 .6 ) as sh o w n by p lo ttin g the
n o rm a lized p o w er ab sorb ed as a fu n ction o f the n orm alized field ina variety o f
granular H T S C sa m p les (T ab le II1.4)
98
-
.
-
■ ------- 1------ “•*
4
P
B
:?
^
2
--------- ------
to 10 p m Y B fc O p w d r ,
36G H z
□
" fr
5 p m B S C C O pw dr, 10G H z
10 p m Y B C O pw dr,
10G H z
-
*
- •
-
■□
•
1
□
__l
0
0
20
•
i
i
60
80
1-------- 1
40
100
T (K )
F ig. 3 .2 2 .
T em perature d ep en d e n c e o f the fie ld param eter H 0 in the c a se o f the
m icron sized p o w d er sa m p les.
99
20
P?
£
78
80
84
82
86
88
T (K )
F ig . 3. 2 3 .
T em perature variation o f the sca le factor a in u nits o f % PN for
10 p.m Y B C O p o w d er sam p le at 10 G H z.
100
90
0.8
to .6
y*.
b
° 0.4
0.2
O.fr
0.0
0.2
0.4
0.6
0.8
1.0
T/T,
F ig. 3 .2 4
T em perature d ep en d e n c e o f o t(T )/a (0 )
Y0 9Ca0 j SrB aC u295 C o c c i '
for the
p ellet at 10 G H z [R ef. 18]. T h e fu ll line
w as ob tain ed from E q .(3 .2 8 ) w ith n = 2 (S IS ) and a ( 0 ) = 0 .2 1 .
101
a (%I?,)
40
20
10
20
30
40
50
60
70
f (GHz)
F ig 3 .2 5 a . T h e freq u en cy d ep en d en ce o f the sca le param eter a
in the c a se o f
the 10 p m Y B C O p o w d er sam p le. T h e full line is a fit to the eq u ation
a = 0 .0 1 7 /( 0 .0 3 + 0 .0 0 0 4 fA2) such that U 0 ) R ~ lOOpV.
20
30
40
f (GHz)
F ig 3 .2 5 b . T h e freq u en cy d ep en d en ce o f the sca le param eter a for the 5 0 p m
Y B C O a g g lo m era te sam p le at 7 7 K. T h e full line is a fit to the eq u ation
a = 8 .5 7 /( 1 3 .3 3 + 0 .0 1 4 5 4 fA2) su ch that I^ O R - 13 0 p V .
103
20
a
(%
PN)
40
10
20
30
40
50
f (GHz)
F ig 3 .2 5 c . T h e freq u en cy d ep en d e n c e o f the sc a le param eter a
for the
5 |im B S C C O p o w d er sam p le at 7 7 K . T h e full line is a fit to the eq uation
a = 0 .0 9 4 /(0 .2 + 0 .0 0 1 l f A2) su ch that I^OJR ~ 6 0 p V .
104
3.4
Universal Virgin C urve - Discussion
T he universality o f the virgin b eh avior se e m s to su g g est that the V C
form s the m o st fu ndam ental co n stitu en t o f the L F IM A p h e n o m en o n . T h e crucial
test o f any m o d e l, th erefore, m ust c o m e from its a b ility to a c c o u n t fo r the V C .
S u rp rsin g ly , in the past, very fe w authors have fo c u ss e d on the V C . In fact, the
V C h as b e e n o ften m ista k en ly id e n tified w ith the n o n -h y ste retic r e sp o n se w h ich
(d iscu ssed in C hapter 4 ) is so m e tim es seen at lo w
W e recall
valu es.
the lin ea rity and re v e r sib le nature o f th e
lo w -f ie ld
m a g n etic iso th erm s o f th ese p o w d er sa m p les (F ig s. 3.1 and 3 .2 ) w h ich stron gly
su g g e s t the sa m p le s to be in the M e issn er p h a se, i.e ., B = 0 w ith in th e grain s. T h e
L F IM A V C , on the oth er hand, e x h ib its an initial sharp rise fo r the sa m e field
region; infact from T ab le 3 .4 w e se e that in m any o f th ese c a se s H 0 , w h ich is the
field required for o n e h a lf saturation absorption, fa lls w e ll w ith in the regim e w here
M is linear in H. For e x a m p le , in the ca se o f the 10 (im Y B C O p o w d er at 4 .2 K ,
linearity in m a g n etic isotherm p ersists up to jt0H - 2 0 m T
(s e e F ig . 3 .1 ) w h erea s
H 0= 4 m T . S o a lso in the ca se o f 5 (im B S C C O p o w d er at 7 7 K , n o n -lin ea rity in
M vs. H cu rve is o b serv e d for ^ 0H > 3 m T (F ig. 3 .2 ), w h ile H 0= 0 .4 m T .
In the
c a s e o f the 5 0 (im Y B C O a g g lo m er a te, th e rev erse is true; w h ile th e m a g n e tic
isoth erm d e v ia te s from linearity for field s as lo w as 1 m T at o n ly 4 .2 K (F ig . 3 .3 ),
the L F IM A V C at 77 K fits to an H 0 as h igh as 5 m T . C learly, there is no sim p le
correlation b etw een the field v alu es o f the tw o resp on ses.
C o n sid e rin g there is no o b v io u s co r resp o n d e n c e b e tw e e n the fie ld
ra n g es in v o lv e d , any m o d el that a s so c ia te s the m ic r o w a v e lo s s to a n o n -z e r o B
in sid e the sp e c im e n is su sp ect. In oth er w ord s, in the c a se s p resen ted a b o v e, use
105
o f critical state m o d els [68, 69] to calcu late the e ffec tiv e B w ithin the sam p le is hard
to ju stify .
T o u n d erstan d th e d ifferen t b e h a v io r in w ea k m a g n etic fie ld s , it
sh o u ld be r e c o g n iz e d that m a g n e tiz a tio n is a m a c r o s c o p ic p ro p erty , i.e . it is
e s s e n tia lly a v er a g ed o v e r the en tire sam p le.
O n the oth er hand, the m ic r o w a v e
a b so rp tion is s e n s itiv e to sa m p le m icrostru ctu re.
If, th erefo re, o n e is g o in g to
co n stru ct a m o d el to ob tain a local B w ith in the sa m p le, the c a se o f ran d om ly
o rien ted p o w d er sa m p le s co u ld turn ou t to b e e x tr e m e ly c o m p lic a te d sin c e o n e
w o u ld h ave to take in to a c c o u n t the p rec ise g e o m e tr ie s o f th e grain s and their
orientation w ith respect to the ap p lied field etc. In fact, o n e m igh t h ave to construct
an en tire h eirarchy o f critical state m o d e ls for the variou s grain sh a p e s p o ssib le .
T h is a sp e c t
o f the p ro b le m w ill b e c o m e m ore ap paren t w h en w e d is c u s s the
h ysteretic resp on se in C hapter IV.
D e sp ite th ese o b v io u s d iffic u ltie s , sev era l [4 7 , 6 2 , 7 0 , 7 1 ] critical
state m o d e ls h a v e b een p ro p o sed w h ere variou s c o m b in a tio n s o f intergrain and
intragrain fie ld s and cu rren t d istrib u tio n s h a v e b een in v o k e d .
A p articu larly
in terestin g critical state m o d el p ro p osed to ex p la in the p h e n o m en o n o f L F IM A is
d ue to Ji e t al. [7 1 , 7 2 ] w h ere th e m ic r o w a v e lo ss is a sso c ia te d w ith the v is c o u s
m o tio n o f flu x o n s a ssu m e d to be present w ith in the sp e c im e n . W e first m ad e an
attem pt to understand the V C data in term s o f this flu x o n m o d el. In the n ext fe w
paragraphs the flu x o n m od el is b riefly introduced. F o llo w in g this w e d em on strate,
taking the particular c a se s o f the 10 |im Y B C O p o w d er, the 5 (im B S C C O p o w d er
106
and the 5 0 p m Y B C O agglom erate for the sake o f argum ent, w h y the m od el fails to
acco u n t for the ob servation s.
In th e flu x o n lo s s m o d e l, the o b se r v e d L F IM A is attribu ted to
d a m p ed flu x o n m o tio n .
F o llo w in g P o rtis e t al, [4 7 ] the n o r m a liz e d su rfa c e
resistan ce for a bulk superconductor can be written as
f i ,( f i ) _ X 0a NS
R
( 3 .7 )
a /2
n
w h ere f represen ts the fraction o f w e a k ly p in n ed flu x o n s (stro n g ly p in n ed flu x o n s
d o not con trib u te to the ab sorp tion) and B the field in sid e the sam p le.
X 0 = / t 0toA
represents the su rface reactance at B = 0 , A b ein g the L ond on penetration depth, a N
th e
n orm al
s ta te
c o n d u c t iv it y
and
8
th e
e le c t r o m a g n e t ic
s k in
d e p th .
B0 = n 0O ) v A 2/<P0 is treated as a m easure o f the e ffe c tiv e fie ld a b o v e w h ich flu x flo w d issip ation d o m in a tes the lo ss p rocess,
v is the flu x o n v is c o s ity and <J>0 the
flu x quantum . It is fruitful to ask h o w w e ll E q. (3 .7 ) d escrib es the o b serv a tio n s.
In t he c a s e o f m i c r o n - s i z e d g r a i ns : F rom F igs. 3.1 and 3 .2 it is clea r that B = 0 for
fie ld s up to about 10 H 0 . T h u s, it is u n lik e ly that a s ig n ific a n t am ou n t o f flu x
r e sid es in the sa m p le.
A straigh tforw ard u se o f Eq. (3 .7 )
is th erefo re hard to
ju stify . If so m e h o w it turns out that the flu x o n m od el is still a p p lica b le o n e runs
in to further d iffic u ltie s .
First, the e x p er im en t c le a r ly s h o w s satu ration at high
fie ld s w h e r e a s E q. (3 .7 ) im p lie s a B'n d ep e n d e n c e for B » B 0 .
S e c o n d , the
n o rm a lized L F IM A (i.e ., a ) redu ces rapidly w ith in creasin g to w h er ea s E q. (3 .7 )
y ie ld s a to~]/2 d e p e n d e n c e at lo w
fie ld s , w h ile for
in d ep en d en t o f freq uency.
107
B »
S o.
f i./f iv
is
a
In t he c a s e o f a g g l o m e r a t e s : T he Z FC isoth erm . F ig. 3 .3 , sh o w s a n on -lin earity in
the M vs. H cu rve for f20H > 0 .8 m T . From this on e co u ld a ssu m e the p resen ce
o f flu x o n s in the sa m p le and g o o d ca u se to ap ply Eq. (3 .7 ).
T he rou g h ly linear
b eh a v io r p red icted b y the flu x o n m od el at lo w ap p lied fie ld s ap p ears to agree
q u a lita tiv e ly w ith th e d ata fo r th e V C (s e e F ig . 3 .1 9 ).
F or a q u a n tita tiv e
c o m p a r iso n , B0 w a s e v a lu a te d for rea so n a b le v a lu e s o f i/ = 10~7 k g /m s, and
X = 10 7m to be = 0.1 T . In other w ord s, the flu x o n m o d el p red icts a linear rise
in R k( B ) / R n fo r 0 < B < 0.1 T. T h is is not seen in the ex p er im en t . Rather, the
linearity in the V C is o b served o n ly for ju 0H < lO m T .
p o w e r a b so rp tio n a p p ro a ch es satu ration .
For (i ^H > 10 m T the V C
T h is is again con trary to the flu x o n
m od el b ecau se if for so m e reason B0 is sm aller, Eq. (3 .4 ) y ie ld s Rs/ R n « B l/2 for
B »
B0 . F in a lly , fo r in c rea sin g CO, the sc a le p aram eter a s h o w s a d ram atic
d ecr ea se w h ile the flu x o n m o d el predicts a co~'/2 d ep en d e n c e at lo w fie ld s and a
freq u en cy-in d ep en d en t R J R N for B » B 0 .
It is clea r from the a b o v e d isc u ssio n that the flu x o n m o d el, albeit at
first sig h t an a ttra ctiv e ap p roach , ca n n o t be u sed to sa tisfa c to r ily e x p la in the
L F IM A V C .
A n im portant c lu e to the true ex p la n a tio n c o m e s from th e fa c t that
the lo w -f ie ld m a g n eto a b so rp tio n is o n ly o b se r v e d in gran u lar sa m p le s.
H igh
q uality thin film s and sin g le crystals do not sh o w sign ifcan t lo ss e x c e p t c lo s e to T c
or for fie ld s > 5 0 m T.
c a u s e s fo r L F IM A .
T h is stron gly su g g e ts that granularity is o n e o f the m ain
F u rtherm ore, o n e o b se r v e s from the z e r o -fie ld m ic r o w a v e
transitions that the m ore granular a sam p le the m ore rounded is the step at the on set
108
o f the m ic r o w a v e transition [com p are F ig 2 .5 and F ig. 3 .9 ],
T h is s u g g e s ts that
on e sh o u ld co n sid e r the sam p le to have a distribution o f T c s. T h is lead s to the idea
that a gran u lar s a m p le is to b e reg a rd ed as a c o lle c t io n o f b o th str o n g ly
su p e r c o n d u c tin g r e g io n s arisin g from the g ra in s and w e a k ly su p e r c o n d u c tin g
region s or areas w h ere su p ercon d u ctivity is d ep ressed . T h e latter m ay c o m e about
due to th e p rese n c e o f d e fe c ts , from n o n -sto ich io m etr ic re g io n s o f the grain s or
from inter- and intra-granular links.
M an y authors [73, 7 4 ] h ave stressed the im p ortan ce o f w eak link s,
in te r - o r in tr a g r a n u la r , to
d e s c r ib e
th e
p r o p e r tie s
o f h ig h
te m p e r a tu r e
su p erco n d u ctin g o x id e s. W eak lin k s h ave a ls o been in v o k ed [74] to acco p u n t for
m icr o w a v e ab sorp tion and m a gn etoab sorp tion . T h e se w eak lin k s h a v e a lso b een
m o d e lle d a s e ith e r J o s e p h s o n ju n c tio n s or as S u p e r c o n d u c tin g Q U a n tu m
Interference D e v ic e s (S Q U ID s).
In R ef. 6 7 , rf-S Q U ID s w ere in vok ed to understand the L F IM A V C
.
D e fe c t s on the grain su rfa ce w e re v is u a liz e d as fo r m in g a su p e r c o n d u c tin g
n etw ork m ad e up o f w e a k ly co u p le d rf-S Q U ID s. A distribution in lo o p areas and
o rien tation to the d c m a g n etic fie ld w a s c h o se n o v er w h ic h the re p o n se o f the rfS Q U I D w a s a v er a g ed .
T h is
m o d el w a s rea so n a b ly s u c c e s s fu l in o b ta in in g a
q u a lita tiv e d escrip tion o f the V C . H o w ev e r, th e d escrip tion [7 5 ] d id not y ie ld the
field s c a le s co m p a ra b le to the ob serv a tio n s. H o w ev e r, the u n iversal nature o f the
V C , w h ic h appears in d ep en d en t o f w h eth er the sa m p le is a sintrered p e lle t or a
p o w d er m aterial or for that m atter the p ro cessin g tech niq ue, q uite clearly ca lls for a
m ore general m od el.
109
W e n o w c o m e to m o d e ls that h ave e m p lo y e d r e sis itiv e ly sh un ted
J o sep h o n ju n c tio n s (R S Js) to represent the w eak link s. G iura e t al. [6 0 , 7 6 , 7 7 ]
p ro p o sed that a gran u lar H T S C sa m p le sh o u ld b e c o n sid e r e d as a n etw o rk o f
in d ep en d en t ju n c tio n s. W h ile at high tem peratures the absorption at lo w dc fie ld
c o u ld b e a c c o u n te d for as due to in d ep en d en t ju n c tio n s, they argue, that b e lo w a
certain J o sep h so n c o u p lin g tem perature 7 ], the lo ss sh o u ld be a sso cia te d w ith the
d ep h a sin g b etw een the ju n ctio n s due to that ap plied field . Furtherm ore, the m od el,
as far as w e k n ow , d o es not y ield an ex p licit freq u en cy d ep en d en ce.
D e sp ite the num ber and variety o f m o d els, to our k n o w le d g e , there
is n o stu d y in the literature that c o v e r s the w id e range o f p h en o m en a d escrib ed in
the se c tio n s a b o v e and attem pts to attribute them all to the e x iste n c e o f w eak links.
A s w e sh a ll s h o w , a rather sim p le m o d el is in d ee d s u c c e s sfu l in rep ro d u cin g
q u a litatively alm ost all o f the experim ental results usin g quite credible valu es for the
r e le v a n t p aran eters.
A n d , a lth o u g h the c o m p le x ity o f the s y s te m m a k e s it a
form idab le task, quantitative com p arison s can be m ade.
T h e RSJ m o d el, first put forward by D u lc ic et al. [1 6 ], is d e v e lo p e d
as fo llo w s . T h e sa m p le is th ou gh t o f as b ein g m ad e o f m an y ran d om ly orien ted
grain s and the w eak lin k s are in vok ed at the su rface o f the sam p le (se e F ig. 3 .2 6 ).
E ach w e a k link is treated as an RSJ. U sin g K ir c h o ffs current law and the v o lta g ep h ase relation for a Josep h son ju n ction (Eq. 1.43) the w ell know n [2 4 ] eq u ation o f
m otion for ac and dc current d iven RSJ is written as
w h ere /
and
r e sp e c tiv e ly .
R are the critical current and the sh u n tin g re sista n c e o f the RSJ
<t>0 = h / 2 e is the flu x q uantum .
C is the ju n ctio n ca p a cita n ce and
co n sid ered ex trem ely sm all sin ce the grain siz e is o n ly o f the order o f a m icron.
/„
is the current in d u ced by the ap p lied dc m a g n etic field H and / nm is the in d u ced
m icro w a v e current o f angular freq u en cy a)mw, and (p is the g a u g e-in v a ria n t p hase
d iffe r e n c e a cross the ju n ctio n (Eq. 1.42). Ign orin g the ca p a cita tiv e term red u ces
Eq. (3 .8 ) to
( 3 .9 )
W e a ssu m e / 0 < 7(,. In the ab sen ce o f m icro w a v es the phase (p w o u ld adjust its e lf
su ch that
/ ( sin (p0 = I0
( 3 .1 0 )
T h e m ic r o w a v e field introd uces o sc illa tio n s o f the p h ase around (p.
W h en both
fie ld s are p resen t and both are very w ea k o n e can w rite (p ~ (p0 + (pmw( 0
lin earize Eq. (3 .9 ) by a ssu m in g (pmH ( t ) «
and
1, i.e .,
sin (p = sin(<p() + (pmJ 0 ) = sin <p0 +
) c o s <p0
( 3 .1 1 )
U sin g E q s. (3 .1 0 ) and (3 .1 1 ) in Eq. (3 .9 ), w e ob tain
1 ^
R2e
A ssu m in g
+ <pmw (/)/,. c o s (p() = Imwe ,w‘
dt
<pmn. - (pamve'"'""' in Eq. (3 .1 2 ), w e get
( 3 .1 2 )
(0
- _________ Llli!__________________________________________ 1-1',
ih(Q„
+ / (. cos<p()
e
2 eR
F rom here on the subscript on the m icr o w a v e current, angular freq u en cy and (p is
n eg lected for co n v en ien ce.
T h erefo re, the p o w er absorbed by each RSJ is g iv en by
(a)
(b)
1 0
I c sincp)<
R
F ig. 3 .2 6 . a) S ch em a tic w eak link in a su p ercon d u ctin g grain, b) A current d riven
r e sistiv e ly sh un ted J o sep h son ju n ctio n circu it w h ere R represen ts the norm al state
resista n ce o f the ju n ctio n and IcSintp is the current through the ju n ctio n . T he
ca p a cita n ce o f the ju n ctio n is ignored.
113
U sin g Eq. (3 .1 3 ) in Eq. (3 .1 4 ) w e g et for the p o w er absorbed by an RSJ
P = P„ — - T
(3 .1 5 !
Pn =
(3 1 6 >
1 + T]
w h ere
j
*LR
is the h ig h fie ld or the n orm al state p o w e r ab sorp tion by the ju n c tio n and the
param eter r) is g iv en by
2
n
(2 el,R \2
=
2
c o s <p0
( 3 .1 7 )
flO)
rj rep resen ts the ratio o f the co u p lin g en erg y o f the J o sep h so n ju n ctio n at a g iv en
tem perature T and dc field /r0H to th e m ic r o w a v e p hoton en er g y , ha).
T h e field
and tem perature d ep en d en ce co m es from the ju n ction critical current It ( H , T ) .
T h e critical current in an RSJ c h a n g es as the extern al dc m agn etic
field is increased. A t any tem perature T the reduction factor due to field is the w ell
k n ow n [24] Fraunhofer d iffraction form u la (Eq. (1 .4 7 ))
( 3 .1 8 )
w h ere O rep resen ts the flu x in the ju n ctio n d u e the ap p lied m a g n etic field .
is the tem perature dependent zero -field critical current o f the RSJ.
1 14
/" (T )
Eq. (3 .1 5 ) y ie ld s the p o w er ab sorb ed by a s in g le RSJ. T o ob tain
the total p o w e r a b so rb ed by the sa m p le o n e h as to r e c o g n iz e that the sam p le
e n c o m p a ss e s several R S Js and sh ould therefore be treated as a c o lle c tio n o f R SJs
w ith a v a re ity o f p a ra m e ter s. F or in sta n c e , th e c r itic a l c u rr en ts, ju n c tio n
resista n ces, ju n ctio n areas and their p rojection s a lon g the ap plied field , A ± , can all
b e e x p e c te d to h ave w id e variations. For sim p licty it is p rop osed to regard rf0 (the
ze ro -fie ld value o f 77 ) as a "lumped" param eter w h ich is co m m o n to all the link s.
T h e net p o w e r a b so rb ed on a p p lica tio n o f fie ld n o r m a liz e d to the h ig h fie ld
a b sorp tion o f the R SJ is then ob ta in ed by in tegratin g the F rau n h ofer d iffra ctio n
fu n ction (E q. 3 .1 8 ) o v e r a d istribu tion o f ju n ctio n areas, i.e.,
^ £ ) = _____ !_____ " 7 _____________
PN
w h ere
~ n mw
<±______
( 3 .2 0 )
„m
l„{l + 7 ? 0 [S>n2n x j ( n x f ]}
I?.R
rj0 = 2 e ——
hco
( 3 .2 1 )
x is related to the e f fe c tiv e ju n ctio n area A L and the su rface fie ld and w h ere the
latter has been set equal to the applied field ( i aH , by
jc =
( 3 .2 2 )
*0
In p rin cip le, o n e sh o u ld in c lu d e the d em a g n etiz a tio n e ffe c ts as w e ll as any field
variation o v e r the ju n c tio n d im e n sio n s.
For sim p lic ity , it is argu ed that th ese
factors can be ab sorb ed in the v a lu es o f the e ffe c tiv e areas.
115
S in c e there is no apriori k n o w le d g e o f the d istribu tion o f e ffe c tiv e
areas, it w as d ecid ed that on e cou ld sim u late the sp ecim en u sin g a flat distribution
P(A )= 1
min
<n<n,
n > n,
0
(3 .2 3 )
so that the average value for the e ffec tiv e junction area is written as
( 3 .2 4 )
and the integration carried out o v er n.
E q. (3 .2 0 ) rep rod u ces th e V C rem arkably
w ell and is sh o w n here in F ig. 3 .2 7 . T he o n ly e x c e p tio n is th e c a se o f th e 5 0 p m
Y B C O a g g lo m er a te sam p le at 4 .2 K w h ere a is large (s e e T a b le 3 .4 ) and the V C
sh o w s a s lig h t fla tte n in g near z e r o -fie ld .
T h is fla tten n in g d o e s n ot c o m e as a
surprise, sin c e after a ll, the ju n ctio n critical current at ex tr e m e ly sm a ll fie ld s d o es
approach zero slop e.
1 16
Normalized Power Absorption
o
o
0
0.
5
10
15
20
DC Field (mT)
F ig. 3 .2 7 .
C a lcu la ted virgin field in d u ced p o w er ab sorp tion u sin g Eq. (3 .2 0 )
w ith n ma x= 2 0
n min = 0 001
17
A rou gh e s tim a te o f r}0 can be o b ta in e d fro m the z e r o -fie ld
m ic r o w a v e ab sorp tion at the tem p eratu re o f interest.
A t 7 7 K th is is ty p ic a lly
arou n d 5% o f the ab sorp tion at T = T* e x c e p t in the c a s e o f the a g g lo m e r a te s
w h er e it w a s fo u n d to be ~ 10% .
T h is im p lie s rj0 v a lu e s b e tw e e n 5 and 3,
re sp e c tiv e ly . For 10 G H z m icr o w a v es this g iv e s for the product l " . R { H K ) valu es
o f 6 0 p V - 1 0 0 p V . T h is valu e co m p a res w e ll w ith the num bers [78, 7 9 ] ob tain ed
for typ ical tunnel ju n ction s.
For com parison w ith exp erim en t it is u sefu l to evalu ate
AP
AP
a
rP N - P
P{x)
Vo
N o te that to g et a sm o o th and m o n o to n ic a lly varyin g A P / a ,
needed.
( 3 .2 5 )
Vo
- n ma > 2 0 is
F or sm a lle r v a lu e s, th e fu n ctio n retain s v e s t ig e s o f th e o sc illla to r y
b eh a v io r o f the F rau n h offer form u la. F or p u rp oses o f co m p u ta tio n nmin = 0 .0 0 1
w a s u sed .
A s a quantitative test o f the RSJ m o d el, an estim ate o f A l w as m ade
as fo llo w s .
F rom E q. (3 .6 ), ^ / x = / / 2 at ^ = H 0 -
U s in g E q. ( 3 ,2 0 ) w ith
fix ed , the valu e o f x (term ed x 0 ) w as ob tain ed by so lv in g
1 \( + Vo ^ 1
( P(xa)
I PN 1+ vl V Vo ) “ 2
i
_
T h u s,
^
_
*0%
(3 .2 6 )
( 3 .2 7 )
1
118
F ig . 3 .2 8 s h o w s
a
()
as a fu n ctio n o f rf{), and o n e n o tes that for r\n in creasin g
b etw e en 1 and 10, x 0 rises by a factor o f 5. Eq. (3 .2 7 ) a llo w s us to interpret the
fie ld p aram eter / / 0 as th e fie ld v a lu e at w h ic h an R SJ o f a v e r a g e area ( A L)
e n c o m p a s se s o n e flu x o n . From Eq. (3 .2 4 ) and
n max - nmin s 2 0 w e see that the
average area (A ± ) s 10A ± and is co n siste n t w ith the v a lu e s ob tain ed for
x 0 . O ne
p oin t o f con cern can be that in F ig 3 .2 8 w e see jc0 -^ co n sta n t as rj()—>0. H o w ev e r,
it sh ould b e noted that r/0 —>>0 w h en / ” —» 0 w h ich occu rs at T ~ T . O n e e x p e c ts
typical ju n ction areas o f A x ~ 2 k d , w h ere d is lik ely to be o f the order o f a fe w p m
(grain s iz e ). A n d b eca u se A in c rea se s rapidly as T a p p ro a ch es T c , {A x ) sh o u ld
a lso b eco m e extrem ely large and w ill ensure that H 0 —y 0 .
U sin g the H 0 v a lu e s listed in T a b le III.4, it is e stim a te d that at 77
K,
A l v a r ie s b e tw e e n
( 0 .0 2 - 1 ) p m 2 , th e s m a lle s t v a lu e s b e in g fo r the
a g g lo m erates and the align ed p ow d er.
S in ce A > 0 .2 p m , the estim ated v alu es for
A x are q u ite reasonable.
119
0.8
0.2
00
F ig. 3 .2 8 . D e p e n d e n c e o f the field param eter x 0 (w h ere the ab sorp tion reach es
o n e -h a lf its saturation v a lu e ) on r\Q, d eriv ed from E qs. (3 .2 1 ) and (3 .2 5 ). T he
s o lid lin e is a linear fit.
120
For further co m p a r iso n s b etw een the data and the m od el o n e sh ou ld look at the
tem perature and freq u en cy d ep e n d e n c e s o f the param eters //,, and a .
W e first
co n sid er the tem perature and frequency d ep en d en ce o f the field param eter
T em p e ra t u r e d e p e n d e n c e a t f i x e d co: E xp erim en tally, it is fou n d that H 0 d ecrea ses
m o n o to n ic a lly w ith in creasin g tem perature, as sh o w n in F ig s. 3 .2 2 a, b. T h is can
be ro u g h ly u n d erstood from the RSJ m o d el as fo llo w s . O ne can w rite Eq. (3 .2 7 )
as
H 0 x X0 j X d . A n d from F ig 3 .2 8 w e s e e that x 0 = a + hrjQ( I ° ) .
So
the
tem perature d e p en d e n c e o f H n c o m e s from both the L o n d o n p en etra tio n dep th
A and the zero -field ju n ction critical current / (°. For th e form er, it is rea so n a b le to
u se the tw o -flu id (Eq. 1.21 ) tem perature d ep en d en ce. For the latter there are tw o
p o s s ib ilitie s .
F o llo w in g A m b eg a o k a r-B a r a to ff, S IS ju n c tio n s h a v e /" ~ ( l - r )
n ear T r w h ile S N S
[2 6 ] req u ire
/ " - ( l - f ) 2. C h o o s in g
an S I S
ty p e o f
tem perature d ep en d en ce for the ju n ction critical current, i.e..
( 3 .2 8 )
o n e obtains
( 3 .2 9 )
w h ere t - T /T . E q. ( 3 .2 9 ) g iv e s a ro u g h ly lin ear tem p eratu re d e p e n d e n c e for
Ho-
F>g- 3 .2 9
s h o w s the e x p e r im e n ta lly o b s e r v e d H 0( T ) / H 0{ 0 ) v e r su s the
reduced tem perature t in the ca se the 10 p m Y B C O p ow d er at
121
1.0
© YBCO peliet @ 10G Hz
A YBCO powder
(10am grains)® 10GHz
0.8
Y«j
0 YBCO powder
(10am grains) @ 36G Hz
.
•
Hq(T)
Hq(0)
0.6
Bi2223 powder
(5pm grains)@ 10G Hz
® \ ^
0.4
• Y (• V
—
0.2
© V &
I
I
0
0.2
I
0.4
0.6
TfTc
F ig . 3 .2 9 a .
Y
I
0.8
1.0
—
T em p eratu re d ep en d e n c e o f the param eter H 0 ; data
are for (i) a Y B C O p e lle t ( 0 ) at 10 G H z, (ii) 10 p m (grain d ia .)
Y B C O p o w d er at 10 G H z (A) and 3 6 G H z ( ) and (iii) 5 p m (grain
d ia .) B S C C O p o w d e r at 10 G H z ( • ) .
/ / 0( 7 ' ) - / / 0(0 )[fl + / 7 ( ( l - r ) ] ( l - r ,*)l/;:
red u ced
te m p e r a tu r e
an d
th e
T h e s o lid lin e is a fit to
w here
p a r a m e te r s
t = T /T c
is
th e
a = 0 and b = I .
1.0
■ YBCOPellet 1(Sample 1-7) + YBCOPellet 2(Sample 1-8) _
0.8
Frequency-10GHz
-
0.6
0.4
0.2
0.4
0.2
F ig . 3 .2 9 b .
0.6
0.8
1.0
T em p eratu re d e p e n d e n c e o f th e p aram eter t f 0 ; data
sh o w n h ere are for Y B C O p ellet 1 (sa m p le II-7, so lid sq u ares) and
Y B C O p e lle t 2 (s a m p le I I-8 , + ) at 10 G H z .
represents
the
H0
values
H 0 = / / ( 0 ) [ a + 6 ( 1 - 0 ] ( 1 - / 4 )1/2
where
p aram eters a = 0 .13 and b = l .
123
T h e s o lid lin e
computed
r = T /T c
using
and
th e
10 G H z and 3 6 G H z, the 5 ftm B S C C O p o w d er at
10 G H z and the Y B C O p ellet
at 10 G H z. In all c a s e s w e see that the p red icted T -variation o f H 0 ob tain ed from
the R SJ
the m o d el
(s o lid lin e ) is in e x c e lle n t a g reem en t.
T h u s, p resen t data
su g g est that the SIS picture is preferable. A sim ilar result w a s ob tain ed by K ish et
al. [4 1 ] w ith regard to a Josep h son ju n ction in sin g le crystal Y B C O .
F r e q u e n c y d e p e n d e n c e a t fi x ed t e m p e r a t u r e : T h e freq u en cy d ep en d en ce o f
H 0 fro m the R SJ m o d el is arrived at as fo llo w s .
F ig . 3 .2 8 in d ic a te s a lin ear
variation o f jc0 w ith r\Q. T h erefo re, from Eq. (3. 2 7 ) w e n o te that th e freq u en cy
d ep en d en ce o f H 0 is c o n tr o lle d by jt0 and h e n c e by 7)0 . E q. ( 3 .2 1 ) in d ica tes that
r\Q is in v ersely proportional to the m icro w a v e freq u en cy. C o n seq u en tly o n e w ou ld
exp ect H 0 to reduce w ith increasing a).
F igu re 3 .3 0 s h o w s the 0) d e p en d e n c e o f
H {) for the 10 fim Y B C O p o w d e r at 7 7 K w h e r e th e s o lid lin e is the R SJ
p rediction.
N e x t, w e m o v e on to the freq u en cy and tem perature d ep en d e n c e o f
the sca le factor a .
Frequency dependence at fix ed temperature:
S in c e 7\ —» 0 b e c a u s e
I(. { H ) —> 0 as field in crea ses, from the RSJ m o d el, the h igh fie ld ab sorp tion for a
s in g le RSJ is w ritten as
1 + 77.)
124
1 + 1h i
¥
1 +
(3.30)
V
K2 e ^ R ,
w h e r e / i s the m icro w a v e freq uency in G H z ( / = ( o f 2 k ).
T h e o b serv e d freq u en cy d ep en d e n c e o f a fo r the 10 Jim
p o w d er sa m p le at 7 7 K is d is p la y e d in F ig. 3 .2 5 a .
YBCO
T h e full lin e rep resen ts the
function
a = ----- ° ' 6
( 3 .3 1 )
1+ 0.02 /
T he o b servation in d eed c o in c id e s w ith the p red iction for a sin g le ju n ctio n . S im ilar
p lo ts h ave b een o b ta in ed fo r the freq u en cy variation o f a i n the c a se o f the 5 (im
B S C C O p o w d er and th e 5 0 |im Y B C O a g g lo m er a te, sh o w n here in F ig s. 3 .2 5 b
and c r e sp e c tiv e ly . In e v e ry c a s e , the fits ind icate that o n e sh ou ld e x p e c t a lo s s o f
n early
50% the lo s s in the norm al state at ze ro fr eq u e n c y .
H o w e v e r , the field
required to obtain th is lo ss w ill be quite h igh as H 0 varies in versely w ith Oi. T h e
o th er in fe r e n c e o n e m a k e s is that at v ery h ig h fr e q u e n c ie s the R S J s d o not
co n trib u te to the lo s s .
It is n o ta b le that o th er e x p e r im e n ts [8 0 , 8 1 ] h a v e a lso
o b serv e d a ~ 0 .1 5 to 0 .3 in the freq u en cy range o f 1 0 -2 0 G H z. A t / =
75 G H z,
and d c m agn etic field < 2 0 m T, W osik et al [82] recorded a saturation absorption o f
< 0 .0 1 w h ich a g rees w e ll w ith ou r ob servation ( a ~ 0 .0 0 1 at / = 6 0 G H z)
125
10
8
6
o
X
£
4
0
Microwave Frequency f (GHz)
F ig. 3 .3 0 . F req u en cy d e p e n d e n c e o f the field param eter H 0 for th e 10 Jim Y B C O
p o w d er at 7 7 K . T h e s o lid lin e is a fit ob tain ed from the RSJ m o d el to the eq u ation
0 . 1 6 + 7 .8 /f.
126
T h e o b serv e d freq u en cy d ep en d e n c e o f a for the 10 |im
p o w d er sa m p le at 7 7 K is d isp la y e d in F ig. 3 .2 5 a .
YBCO
T h e fu ll lin e rep resen ts the
function
a = ----- Q;6
1+ 0 .02 /
,
( 3 .3 1 )
T h e ob servation in d eed c o in c id e s w ith the prediction for a sin g le ju n ctio n . S im ilar
p lo ts h ave been ob tain ed for the freq u en cy variation o f a in the ca se o f the 5 p m
B S C C O p o w d er and the 5 0 p m Y B C O a g g lo m er a te, sh o w n here in F ig s. 3 .2 5 b
and c r e sp e c tiv e ly . In ev ery c a se , the fits ind icate that o n e sh o u ld ex p e c t a lo ss o f
n early 50% com p ared to the lo ss in the norm al state at zero freq u en cy. H o w ev e r,
the field required to ob tain this lo ss varies in v e rsely w ith to . T h e oth er in feren ce
o n e m a k es is that at very h igh freq u e n c ie s the R S Js d o not p articip ate in the lo ss
m e ch a n ism . It is n o ta b le that oth er ex p er im en ts [8 0 , 8 1 ] h a v e a ls o o b serv e d a ~
0 .1 5 to 0 .3 in the freq u en cy range o f 1 0 -2 0 G H z. A t / = 7 5 G H z, and d c m agn etic
fie ld < 2 0 m T , W o s ik et al [8 2 ] record ed a saturation ab sorp tion o f < 0 .0 1 w h ich
agrees w ell w ith our ob servation ( a ~ 0 .0 0 1 at / = 6 0 G H z)
T o ob tain the V C from the RSJ m od el o n e n eed ed to integrate the
p o w e r a b so rb ed by a s in g le ju n c tio n o v e r a d istrib u tion o f ju n c tio n areas.
Eq.
( 3 .3 0 ), h o w e v e r , is the satuartion p o w e r ab sorb ed by o n e R SJ. It is by n o m ean s
straigh tforw ard to g o from o n e RSJ to an actual sam p le w h ich w ill h ave a m esh o f
R S J s w ith a variety o f 7° and R .
O n e can m ak e so m e p ro g ress by n o tin g that
m an y in d ep en d en t m e a su r em en ts [7 8 , 7 9 ] su g g e s t that
YBCO.
7(°7? =
lO O pV at 7 7
K
in
If so , E q. ( 3 .3 1 ) can be rea lized from Eq. (3 .3 0 ) by in v o k in g a s in g le
127
value o f rj() for e v e ry RSJ and u sin g a su itable p rop ortion ality factor to sim u late a
sa m p le. That is, o n e w rites.
0 .0 1 8
0 .0 3 + 5 x 1 0 ^ / "
to con form to I ° R N ~ 1 0 0 /tV .
T e m p e r a t u r e d e p e n d e n c e a t f i x e d f r e q u e n c y . From Eq. (3 .3 0 ) it is clea r that the T d ep e n d e n c e o f a , n o rm a lize d to the zero -k elv in v a l u e a ( 0 ) , w ill c o m e from the
tem perature variation o f the zero-field junction critical current /° . That is a ( t ) m ay
be ex p ec ted to fo llo w
“ ( ,) -
1
a(0 ) l +c(l-r)"
( 3 .2 8 )
w h ere y w ill b e c o n tr o lle d by w h eth er the RSJs are regarded as S IS ( y = l ) or S N S
( y = 2 ) in ch aracter.
p o w d e r sa m p le ( T
F ig . 3 .3 0 a s h o w s a ( t ) fo r the c a s e o f th e 10 [im Y B C O
= 8 9 K ) w h ere y = 2 w a s u sed .
H o w e v e r , g iv e n that the
varaition in a is m o st sig n ific a n t in the region v ery c lo s e to T c o n ly and g iv e n the
ex p erim en ta l errors in b oth T and a o n e can n ot m ak e a clea r d istin ctio n b etw e en
S IS and S N S ju n c tio n s b a sed on or(/) alon e. In fact o n e co u ld u se y = l , i.e ., an
S IS fit fo r th e sa m e data if o n e is w illin g to use a slig h tly d ifferen t T c as sh o w n in
F ig . 3 .3 0 b . In th e c a s e o f Y o .9 C a o . 1 S r B a C u 2 . 9 5 C o o . 0 5 0 7 - y ( T ( - 8 0 K )
p a ra llelp ip ed cera m ic sa m p le, ob tained from R. Suryanarayanan, it w as fou n d that
a ( r ) w a s b etter d e scr ib ed by S IS ju n ctio n s (s e e F ig .3 .2 4 ). In th e c a s e o f cera m ic
B S C C O sa m p le s [6 5 ] the S N S ju n ctio n s w ere fou n d m ore favorab le in d escrib in g
a ( t ) . T h e m o re im p ortan t o b serv a tio n is to n ote that
128
a is ro u g h ly a co n sta n t
b e lo w t< 0 .6 T c su g g e stin g that the num ber o f R SJs con trib u tin g to the lo s s d o e s
not ch a n ge drastically with temperature.
129
78
80
82
84
86
88
T (K )
F ig. 3 .3 0 a . T em p eratu re d ep en d e n c e o f tx(T)/<x(0) the 10 |im Y B C O p o w d er
at 10 G H z. T h e so lid lin e w a s ob tain ed from Eq. (3 .2 8 ) w ith y = 1 (S IS ) and
param eters a ( 0 ) = 2 3 .5 and T c = 8 8 .5 K.
130
©
8
78
80
82
34
86
88
T ( K)
F ig. 3 .3 0 b . T em perature d ep en d e n c e o f <x{T)/<x(0) the 10 p m Y B C O p o w d er
at 10 G H z. T h e so lid line w as ob tain ed from Eq. (3 . 2 8 ) w ith y = 4 (S N S ) and
param eters o t(0 )= 2 2 .4 and T t= 8 9 .5 K . N o tic e that both S IS as w e ll as S N S
type o f tem perature d ep en d e n c e fit the data eq u a lly w e ll if o n e a llo w s for a
slig h t c h a n g e in T .
Chapter IV
Hysteresis
It is n o w w e ll e sta b lish e d [7, 4 3 , 8 3 , 8 4 ] that
su b seq u en t to the
o b serv a tio n s o f the V C , if a granular sa m p le is e x p o s e d to a c y c lic d c m agn etic
field varied linearly b etw een the m axim al field s, ± / i 0/ / max, o n e o b ta in s h y ster esis in
the L F IM A . In other w ord s, the sign al is k n ow n to be d ep en d en t on the m agn etic
h isto ry o f the sa m p le.
F or e x a m p le , c o o lin g the sa m p le in a fie ld p ro d u ces an
o ffse t along the field -a x is o f h ysteretic lo o p equal to the c o o lin g field . A ls o the loop
structure is d ep en d en t on the m axim u m fie ld sw e e p . A lth o u g h the latter a sp ect is
gen era lly agreed upon, there has b een no sy stem a tic stu d y, e sp e c ia lly as a function
o f tem p eratu re and fr e q u e n c y .
p h en o m en on as a fu n ction o f
H ere w e p resen t a s y s te m a tic stu d y o f th is
T and CO.
T h ro u g h o u t th is in v e stig a tio n w e h a v e fo c u s s e d on th e 10 ^im
Y B C O p ow d er sam p le sin ce the sig n a ls at m ost tem peratures and fr eq u en cies w ere
largest o f any sam p le. S o m e m easu rem en ts on the 5 |im B S C C O p o w d er and the
5 0 [tm Y B C O a g g lo m er a te h ave a lso b een m ad e.
M e a su r em e n ts on a separate
B S C C O p e lle t w ere carried out by S esh u B ai et al [6 5 ] and sim ila r trends w ere
o b se r v e d .
T h e resu lts o f th e p resen t in v e stig a tio n are p resen ted in se c tio n 4.1
f o llo w e d by a d is c u s s io n
in s e c tio n 4 .2 .
It is sh o w n that th e h y s te r e s is
p h en o m en o n can be a cc o m o d a ted by a sim p le e x te n sio n to the RSJ m o d el w h ere
the sa m p le is c o n sid e r e d to be an e n s e m b le o f R S Js p la c ed in an en v iro n m en t
w h ere the ap p lied dc am g n etic field is a u gm en ted by random fie ld s gen erated by
flux trapped at random sites in the sp ecim en .
132
4.1
Results
A sch em atic o f m agn etoabsorp tion data sh o w in g a typical h ysteresis
lo o p is sh o w n in F ig . 4 .1 .
T he ch aracteristic p aram eters d e fin in g the h y ster sis
lo o p are 5 H, the fie ld separation b etw een the m in im a, AP ^ , the ab sorp tion at the
m in im a and AP IRA, the irreversib le rem anent ab sorp tion for zero fie ld , all o f w h ich
are fu n ction s o f H max .
A t this point it is im portant to n ote that the h y ster esis lo o p s are a lso
affected sligh tly by the field sw eep rate
A sy stem a tic study o f S H for v ariou s
dt
s w e e p rates in d ic a te s that a fa ste r s w e e p rate e n h a n c e s
d ep en d en ce on
S H.
H o w e v e r , the
is m u ch stronger. T h is is represen ted p icto ria lly for the ca se
o f the 10 Jim Y B C O p o w d e r at 10 G H z and 4 .2 K in F ig . 4 .2 , w h ich s h o w s a
m o n o to n ic in crease in
o b serv ed
S H w ith in crea se in th e s w e e p rate. A ls o m arked are the
S H valu es for the for the m axim u m field sw e e p ( / t 0/ / max) ap p lied . W h ile
the en h an cem en t in
S H is o n ly about 0 .2 m T for ev e ry 0 .5 m T /s in crease in sw e e p
rate, w e n ote that k eep in g the sw e e p rate fix e d and in crea sin g
sig n ific a n tly larger in crease in 8 , r
resu lts in a
F or e x a m p le , from F ig . 4 .2 , w e se e that for
= 18 m T , S H is as high as 0 .3 m T . H o w ev e r, g o in g by the linear in crease
in
S H w ith the rate o f s w e e p , it w o u ld be n ece ssa ry to sw e e p the field at as m uch
as 1.6 m T /s to obtain S H ~ 0 .3 m T.
A t th e fa ste s t s w e e p rate u se d h ere, 0 .9 m T /s, th e s w e e p rate
d e p e n d e n c e m ay lea d to an o v er estim a tio n o f 8 U by < 5 % .
T h e sh a p e o f the
h y ste r e sis lo o p w a s a lso verified as a fu n ctio n o f the s w e e p fu n ctio n u sed for the
dc fie ld .
C h a n g in g from a trian gu lar w a v e fu n c tio n to a sin u so id a l o n e had
n eg lig ib le e ffe c t on the lo o p structure.
133
>•>
-C
‘a.
o
>>
£u
</>
OJQ
£
134
the VC.
—
gO
The dc field is swept between
±/*0H niai.
sam ple. The dashed
cs:
u
u-
in a granular HTSC
line represents
2
aj
C
?
O
o
microwave absorption
[arb. units]
AP(H)
•a
CJ
a
"3
0.8
22.1 mT
0.6
20.6 mT
0.4
P
18.9 mT
18.3 mT
17.2 mT-
-§
0.2
X
17 mT
15.5 m T
to
0.0
-
0.2
-0.4
2
0
dc field sweep-rate (mT/s)
F ig . 4 .2 . C o m p a riso n b etw e en the e ffe c t o f dc field sw eep -ra te and th e m a x im a l
fie ld |i.()Hmax on the h y ster esis loop structure for the 10p .m Y B C O p o w d er at
10 G H z and 4 .2 K . T h e o p en squares are for a m axim al fie ld o f 11 m T . T he
so lid c ir c le s in d ica te § H for th e corresp on d in g m a x im a l fie ld m arked.
135
For a co m p lete description o f the hysteretic L F IM A , it is essen tial to
d efin e a tem perature dependent threshold value ( H Th) for A/max, T h e o b serv a tio n s
fall into the fo llo w in g three categories.
(i) R e g i o n R + : occu rs for H > H n and is sc h e m a tic a lly sh o w n in F ig . 4 .3 a. It
is o b se r v e d in all sa m p les. N o tin g the d irection o f the field s w e e p , it is clear that
the a b sorp tion during sw e e p d ow n is low er than that for sw e e p up. A ls o there are
sizea b le p ortion s o f the loop w here AP is le ss than that on the V C .
AP mn o cc u r s
on the sam e sid e as H mM and this is indicated by sayin g that 8 „ is p o s itiv e . It w a s
d isc o v e r e d that R +
is term in ated o n the lo w H max en d b y H Th. T hat is, o n ly for
/ / rTiax > H Th, is 8 H s iz e a b le and p o s itiv e .
w ith
F igu re 4 .4 s h o w s th e variation o f S H
/ i 0/ / max for the ca se o f th e 10 |xm Y B C O p o w d er at 10 G H z fo r th ree
d iffer en t tem p eratu re settin g s. N o te that at lo w e r tem p eratu res a h ig h er fie ld is
required for the o n set o f region R + , in oth er w ords, H n in creases w ith d ecreasin g
tem perature. T he effec t o f the m axim al field fX0H ^
on the lo o p structure, i.e ., 8 H
and A / ^ , at fix e d tem perature for a typ ical ca se (1 0 |im Y B C O p o w d er sam p le at
10 G H z and 5 0 K ) can b e seen in F ig. 4 .5 .
R e g i o n Ro '■A t the th reshold field , i.e. H = H Th, the h y ster esis lo o p c o lla p s e s and
8 h = 0 . For e x a m p le , a typ ical lo o p in this region is sh o w n in F ig. 4 .3 b fo r the 10
|im Y B C O p o w d er at 10 G H z and 7 7 K . T h is region is a lso o b se r v e d in all the
s a m p le s .
8 lf = 0 , and th e sig n a l is r e v e r sib le o v e r m o st o f the fie ld s w e e p .
H o w e v e r , for fie ld s c lo s e to zero, AP is high er than that on the V C . T hu s there is
an irreversible rem anent absorption t±PIRA- R ed u ction o f / / mai in this region ca u ses
AP, RA to drop (for e x a m p le , F ig. 4 .6 , 10 p m Y B C O p ow d er at 10 G H z and 7 7 K)
but not n e c e ssa r ily van ish , as H imx —» 0.
In so m e sa m p le s R q b e h a v io r p ersists
d o w n to rather lo w H max. S in ce the A P v s . H cu rve is rather c lo s e to the V C , other
136
authors in the past [62] have analysed such data as if they w ere the sam e as the VC.
T he n on-zero valu e o f AP !RA clearly su g g ests that this is not ju stified . In particular,
w h ile (
]
= 0 for H —» 0 , this is not true for the V C .
) rh
R e g i o n R _ : S ch em a tic a lly sh o w n here in Fig. 4 .3 c , occu rs for H mm < H n and is
m arked by a n eg a tiv e S H. P re v io u sly
[4 3 ] h y ster esis w ith su ch a reversed lo o p
structure w a s o b serv e d at 3 6 G H z. H o w ev e r, in that ca se , it w as later co n fir m e d
to be an artifact brought about by the p resen ce o f stray fie ld s in the c a v ity . Care
w as taken in this stu d y to a v o id su ch sy stem a tic errors and the reversed h y ster esis
lo o p structure v erified repeated ly. T h e n eg a tiv e S H v a lu e s w ere fou n d b oth at 10
and 3 6 G H z in the ca se o f the 10 |xm Y B C O p o w d er sam p le.
Fig, 4 .7 s h o w s S H
as a fu n ction o f H mm for th is sa m p le at 4 .2 K and 3 6 G H z.
In co m p a r in g w ith
F ig . 4 .4 o n e n o te s that th e th re sh o ld fie ld s h o w s little c h a n g e w ith fr e q u e n c y
rem ain in g at 14 m T at both 10 G H z and 3 6 G H z and as b efore d e fin e s the lo w er
bound f o r /? + .
< 0 .2 m T .
U n fo rtu n a tely , at lo w
the m a g n itu d e o f 5 „ is q u ite sm a ll,
H o w e v e r , th e cr o ss o v e r from R + to R _ is q u ite clear.
T h at is, the
L F IM A for u p -sw e e p n o w lie s b e l o w that for the d o w n -sw e e p and AP mn occu rs
after field reversal. In the p resent exp reim en ts this w as seen on ly in 10 jxm Y B C O
p o w d er
and th e B S C C O p o w d er at T = 4 .2 K .
H o w e v e r , S esh u B a i et al [6 5 ]
have o b served a n egative S H in sintered B S C C O sa m p les at as high as 7 7 K. Such
a reversal in lo o p structure h as a lso been co n firm ed in a sin tered Y B C O sa m p le
[8 5 ],
137
AP{H)
units]
[arD.
mm
m ax
1 2 .0
max
0
H0H — ►
F ig . 4 .3 a .
H y steretic L F IM A in the region R+ for the 10 (im Y B C O p o w d er at 10
G H z , 4 .2 K. T h e arrow s rep resen t the d irectio n o f s w e e p o f the m a g n etic field .
N o te that the p o w e r ab sorb ed on d o w n sw e e p is lo w er than the p o w e r ab sorbed for
up sw e e p . A ls o S H is taken as p o sitiv e. T he d ashed cu rve represen ts the V C .
138
o
O
<L>
CL
U
c
4»
-o
C
u
a.
u
-o
2
o
E
o
xi
r*i
5?
E
139
Note that S H~ 0 but a non-zero
o
oh
o
as
field.
c
AP,RA rem ains.
of sweep
o
the direction
of applied
E
Arrows indicate
u
QQ
>■
powder at 10 GHz, 77 K.
o
140
VC has been omitted in order to avoid any con fu sion.
/I
t—
E
CO
X
:£
n e
■s
-C
c§
E
CO
o
w
LO
Xo
3.
u
u
c
u
-a
CD
a
u
Cl
W
T3
C
Cl
O
O
O
CO
in
O
O
O
co
O
CM
I-
E
X
c<0
141
/?+.
2
H
LE
field
't
threshold
1/5
of region
E
O
o
o
So
a
<
uu
for the appearance
9^ -
Error
in
X c\l
/ioHjf, ~ ± lm T .
CO
do
iE
10 pm YBCO powder @ 10 GHz, 50 K
VC
AP(H)
[arb. units]
^ o Himax
21 mT
17 mT
14 mT
11.5 mT
9.3 mT
0
10.5
21
p 0 H (mT)
F ig . 4 ,5 . E ffe c t o f the m axim al field ( | t 0H majl) on the h y ster esis lo o p structure, in
p articular, A P mB, the m in im u m in the ab sorp tion and the ab sorp tion on return to
zero fie ld , AP IRA fo r the 10 |im Y B C O p o w d er at 10 G H z, 5 0 K.
142
1
------1--------- ---------- 1----
*
□
1.2 -
-
Vi
1.0
□
-
X) 0.8 -
-
□
c3
0.6 -
-
-
□
0.4
-
0.2 -
-
□
0.0
i
3
.
i
4
5
max (m T )
F ig. 4 .6 .
T h e irreversib le residual ab sorp tion as a fu n ction o f the m ax im u m
a p p lied field in the c a s e o f the 10 (im Y B C O p o w d er at 7 7 K and 10 G H z.
Error in AP
IRA
is about 10% o f the o b serv e d valu e. Error in fie ld is a lso
about 10% o f the value.
143
1,0
10pm YBCO powder at 36GHz, 4.2K
0.8
0.6
(1LU) H'
«o
0.4
0 .2
0
-0 .2
-0.4
0
2
4
6
8
10
12
14
M-O^max (rn"0
16
18
20
22
24
F ig. 4 .7 . T h e fie ld separation S h o f the m in im a in the h y steresis lo o p as a function
in increasin g H miiX v a lu e s in the c a se o f the 10 p m Y B C O p o w d er at 3 6 G H z, 4 .2
K.
N o te that 8 H is sm a ll but n eg a tiv e at H c lo s e to ze ro co r r e sp o n d in g to the
reg io n R _ , then rem ain s c lo s e to ze ro (r eg io n R^) and fin a lly s h o w s a gradual
increase (region R + ) as the field span 7 /mai is increased .
14 4
0.8
0.6
H
a
X
to
0.0
-
0.2
-0.4
0
5
10
15
20
25
^ 0 H m a x (m T )
F ig . 4 .8 .
S H versu s the m a x im a l fie ld for the 10 (im Y B C O p o w d er at 10 G H z ,
4 .2 K.
N o te that th e th resh old fie ld , i.e ., th e field required for th e o n se t o f the
r e g io n
R+
is a b o u t th e sa m e as at th e h ig h e r fr e q u e n c y ( s e e F ig . 4 .7 ) .
145
W ith o n e n otab le ex c ep tio n [86] w here m easurem en ts w ere d on e on
sp e c ia lly prepared grainy (0 .5 p m ) film s, the ch a n g e in g o in g from R + to R _ , i.e.,
a reversal in the lo o p ch aracteristics d u e to a ch an ge in field am p litud e a lo n e , d o es
n ot appear to h a v e b een stu d ied sy ste m a tic a lly in p rev io u s reports on the L F IM A
p h en o m en o n .
T h e th resh o ld fie ld , H n , w a s m ea su red eith er by o b se r v in g the
h y s te r e s is lo o p at h igh
/ / max and lo w e r in g H msa
u n til th e lo o p c lo s e d or by
starting at a lo w A/max value and increasin g it until a p o sitiv e d H appeared. T he tw o
va lu es thus ob tain ed agree to w ith in the exp erim en tal error. It m ust be p oin ted out
here that errors in f i nH n are as h ig h as ± 1 m T b eca u se ( as can be see n in F ig.
4 .3 ) the S H vs.
cu rve ap proach es the field axis tangentially.
A sy ste m a tic stu d y o f th reshold field w ith T in d icated that at fix ed
T,
H n red u ces o n ly slig h tly w ith in c rea sin g CO.
On th e oth er hand , at fix ed
freq u en cy , H n e x h ib its a stron g tem perature d ep en d e n c e. T he T -d e p e n d e n c e o f
H n , at 10 G H z , for tw o m icro n s iz e p o w d ers, Y B C O and B S C C O , and the 5 0
p m Y B C O a g g lo m er a te is sh o w n is sh o w n in F ig. 4 .9 . O n e n o te s that in ev ery
case
in crea ses lin early w ith d ecr ea sin g T and rou gh ly f i 0H n (4 .2 K ) = 10
m T , T a b le. IV . 1 lists the v a lu e s record ed o f f i 0H n at d ifferen t tem p eratu res for
the variou s sa m p les.
146
Table. IV. 1.
Threshold field values as a function of sample
temperature at 10 GHz.
T(K)
S a m p les
10 p m Y B C O p ow d er
2 p m Y B C O p ow d er
50 pm Y BC O
77"'(K )
89
89
92
H0H n ( m T )
it
4 .2
14
18
11.5
50
6 .6
77
3 .9
4 .2
9 .1
30
4 .6
50
3 .3
77
2
4 .2
9 .2
50
4 .8
77
2 .8
18
1 0 .2
50
6 .6
77
4 .7
4 .2
4 .5
54
2 .8
59
2
81
1.2
agglom erate
88 pm Y BC O
87
agglom erate
5 p m B S C C O p ow d er
105
# Error in ju „ H Th ~ ± lm T
147
★
1.0
V
VV
0.8
H jh CO
H Th (0)
0
5pm BSCCO powder
10pm YBCO powder
A 50pm YBCO agglomerate
(•A
0.6
0.4
—
0.2
0
i
i
i
i
\i
0.2
0.4
0.6
0.8
1.0
T/Tc — ►
F ig . 4 .9 .
T em p er a tu re v a ria tio n o f th re sh o ld fie ld f i 0H n .
The
s o lid
lin e
rep resen ts H n ( T ) - { J L)m A . T h e data are for (a) 10 p m Y B C O p o w d er ( O ) , (b)
5 0 p m Y B C O a g g lo m e r a te (A ) and (c) 5 p m B S C C O p o w d er (★ ) , all at 10 G H z.
148
4.2
Discussion
T h r e s h o l d f i e l d : T he e x iste n c e o f a threshold field for the on set o f h y steresis w as,
at first, th o u g h t to be tied to the lo w er critical field , H r t.
A n d a lth o u g h
one
o b se r v e s no d ev ia tio n from linearity in the m agn etic isoth erm at com p arab le field
v a lu e s (se e F ig s. 3.1 and 3 .2 ), the m agn itu d e o f H Th is not to o d ifferen t from H t]
[87] ex p ected for th ese m aterials esp ec ia lly w hen allow an ce is m ade for the fact that
the su rfaces are far from sm ooth .
H (l in th e c a s e o f c o n v e n tio n a l su p erco n d u cto rs sh o u ld vary as
l/A 2 y ield in g a T d ep en d en ce o f
m a g n e t iz a tio n
d a ta
su ggest
T
I-
that
, w ith in the tw o -flu id m o d el. T h e dc
th e
pow ders
behave
as
c o n v e n t io n a l
su p ercon d u ctors. H o w e v e r , the o b serv e d tem perature variation sh o w n in F ig. 4 .9 ,
is m ore linear and u n lik e the m uch slo w er T d ep en d en ce ex p e c te d for con v en tio n a l
su percon d uctors. It sh ou ld , n ev erth eless, be p oin ted that there are so m e theoretical
rea so n s [8 8 ] to b e lie v e that H cl in la y ered su p erco n d u cto rs su ch as the H T S C
o x id e s , sh o u ld s c a le linearly w ith T. S u ch a linear d ecrease has a lso been reported
in
R e f.
87.
A
lin e a r
ch an ge
in
A c o u ld a ls o be a ttrib u ted to d -w a v e
su p erco n d u ctiv ity [8 9 ], if that is appropriate to H T SC .
A n oth er p o ssib ility is that H n is a v e stig e o f the B ea n -L iv in g sto n
[9 0 ] su rface barrier field H s . H o w e v e r , the fo llo w in g tw o p o in ts d o not support
su ch a c o m p a r is o n .
g iv en by the relation
F irst, in the c a s e o f H T S C o x id e s , w h ere % «
A,
H s is
w here £ is the c o h e r e n c e length. From Eq. 4.1 w e see that the v a lu e s o f H s is o f
Ht
th e o rd er o f th e th e r m o d y n a m ic f ie ld
and m u ch
g r e a te r
th an
H r
N e v erth ele ss, co n sid erin g that the sam p les are p ow d ers m ade o f grains o f arbitrary
sh a p e , w h o s e su r fa c e s are not n e c e ss a r ily s m o o th , lo ca l e n h a n c e m e n ts o f the
ap p lied field is very lik ely and co n seq u en tly flu x c o u ld enter at co n sid era b ly low er
field s. T h e seco n d argum ent against id en tify in g HTh as
H s is that from E q. 4 .1 ,
u sin g the tw o -flu id m o d el for A , o n e ob tain s a tem perature variation o f the form
(l - (7 ’/7 't )2 J w h ile the ob servation s indicate a linear T d ep en d en ce for Hn .
A n oth er p o ssib le exp lanation is that H n is d u e to the fin ite critical
current o f the J o sep h so n ju n ctio n . In this c a s e H n = J c ( H Th )A . I f w e c h o s e the
typical em pirical T d ep en d en ce o f Jt. for an S IS ju n c tio n , i.e .,
then o n e ob tain s the so lid line sh ow n in F ig. 4 .9 as a fit to the o b serv e d T variation
of
H n r e d u c e d to its z e r o -k e lv in v a lu e .
N o te that th e a g r e e m e n t is q u ite
sa tisfactory.
L o o p A n a l y s i s : W e have asserted b efore that the u n iversality o f the V C represents
the fu ndam ental L F IM A resp on se. I f so, it fo llo w s that h y steresis h ap p en s w h en a
p rev io u sly e x p o s e d sa m p le se n se s a field w h ich is no lo n g er eq u al to the ap p lied
fie ld H, th e d iffe r e n c e b e in g c o n tr o lle d by the m a g n e tic h isto r y .
d escrib e h ysteretic L F IM A on e should write
150
T h at is, to
AP { H )
(4.2)
a
F o rm a lly .
H eJ} = H + H
. A n o n - z e r o H rf; is required b eca u se,
in the h ysteretic state, A P ^ O for all H . A rg u a b ly , the local fie ld , H I0C arises
from flu x (o r attendant current lo o p s) random ly trapped in the grain surface g iv in g
rise to d ip olar fie ld s. A first p rin cip les ca lcu la tio n to extract H L0C is, n e e d le s s to
sa y , hard to form u late.
In stead , to o b ta in s o m e id e a o f H L0C, o n e c o m p a r e s h y ste r e tic
c u r v e s su ch as sh o w n in F ig s. 4 .3 (a ), (b ) and (c ) w ith their c o r resp o n d in g V C s.
It turns ou t that in the r e g io n s R q and R_ the Hl0C requ ired is very sm a ll in
c o m p a r iso n w ith that in the re g io n R + w h er e h y s te r e s is is m o st p r o n o u n c e d .
T h erefo re, w e first concentrate on the h y ster esis in region R+. A s an e x a m p le , Fig.
4 .1 0 sh o w s the variation o f H U)C as the m agn etic field is sw ep t from 21 m T to -21
m T and back to 21 m T for the 10 jxm Y B C O p ow d er sa m p le at 7 7 K and 10 G H z.
It is n o ta b le that at large H , w h e n (d H f d t ) c h a n g e s s ig n , an in c r e a sin g H ^
o p p o s e s / / , resu ltin g in Heff< H . C lo s e to zero field , H LOC reverses d irection and
now
Heff> H . S u r p r isin g ly , o v e r th e re m a in in g part o f the risin g fie ld s w e e p
H l o c is c lo s e to zero. T h is is clea rly e v id e n t in F ig. 4.11 w h ich s h o w s o n e h a lf
c y c le o f a typ ical R+ h y steresis loop . T h e V C is represented on eith er sid e by the
d a sh e d cu rv e s and has b een in clu d ed for e a s e o f com p arison . A s has b een already
p o in te d o u t, w h e n fie ld is r e d u c e d th e h y ste r e tic a b so r p tio n g r o w s to be
sig n ific a n tly lo w e r than that on the V C but on field reversal the h y ste r e sis cu rve
e s s e n tia lly o v e r la p s th e V C b e v o n d a fe w m T . T h is fin d in g p r o v id e s stron g a
p o s t e r i o r i support to the claim that in d escrib in g the V C on e n eed s to co n sid er o n ly
the applied field . N e x t, the fact that H l0C is con trolled by / / max can be clea rly seen
in F ig . 4. 11 w h ere H w c extracted from the h y ster esis lo o p s has been p lotted as a
function o f the applied field for three different
v a lu e s in the c a se o f the 10
Jim Y B C O p o w d er at 10 G H z and 77 K. F ig. 4 .1 2 is a sim ilar p lot for the sam e
sa m p le at 5 0 K. N o tic e that, in both c a se s the am p litu d e o f H LOC is greater for
larger
valu es and, as d escrib ed a b ove, d ecrea ses w ith redu cin g ap p lied field
ch a n g in g sign c lo s e to zero field . H LOC fo llo w in g field reversal b e c o m e s n eg lig ib le
in both c a se s n o matter w hat the m axim al o f the field sw e e p used.
O n c e a g a in , sev e ra l m o d e ls h a v e b een p ro p o sed to e x p la in th e
h y steretic lo o p and m ost o f th ese use the B ean m od el [2 7 ] in v o k in g eith er or both
intergranular and intragranular trapped flu x. A lth o u g h this m ay b e v a lid for so m e
ex p er im en ts, o n e m u st k eep in m in d that in the m ic r o n -siz e d p o w d ers, the static
m a g n e tic r e s p o n se (F ig . 3.1 and 3 .2 ) is e s s e n tia lly r e v e r s ib le .
T h e r e fo r e a
straigh tforw ard u se o f c a lcu la tio n s b ased on full fie ld penetration in to the grain is
far from ju stifie d . M ore im portatntly, h o w ev e r, it h as to b e r e c o g n iz e d that th ese
are p o w d ers, c o m p o s e d o f grain s w h o se p rec ise g eo m etr y is u n k n o w n and to add
to the c o m p le x ity , can h a v e th e ab sorb in g sites ran d om ly lo ca te d o n its su rface.
If, th erefore, o n e p u rsues in d e v e lo p in g a critical state m o d el then the d escrip tion
w o u ld n ot o n ly require a w h o le array o f m o d e ls to ob tain the g e o m e tr y o f the
trapped flu x but a lso take into con sideration the random p lacem en t o f th e absorbing
s ite s (th e R S J s).
U n d o u b te d ly , su ch a d e sc r ip tio n is e x tr e m e ly c o m p lic a te d .
In ste a d , it turns o u t, that o n e can g a in fu rth er in sig h t in to th e h y s te r e s is
p h en o m en o n by ex ten d in g the RSJ m o d el, presented in C hapter III, as fo llo w s.
152
CM
CO
c
O
C
—
"3
a
j
2
3
a
c_>
'o o
a;
<L>
YBCO powder (grain diameter-1 O^m) at 10GHz, 77K
o
4>
o
CJ
CO
X
2
CO
0J
■o
■
o
ju
cl
CL
:0
<0
CM
-£
v;
3
in
O
CM
c.
>
CO
a:
CM
o
=3.
23
Oj
00
T3
8
«
JO
H
O
u
s
1/5
&b
£
CM
CM
o
153
O
c/5
X
U
r~
r■o
c
CM
N
X
a
©
?5
1_
V
-o
o
CL
o
u
CQ
>E
o
00
CO
Cl.
CJ
CM
4
>
C
w
Cj
■C
H
-X
o
c0
£>
•a
c
CO
H
E
CM
CM
o
"
f-
m
CT
LU
60
c
E
CM
CM
T>
<U
+
E
°
i
-S
°
m
'*■>
c3
°
c
1
—'
:c
'■£ =L
5 --------
3 s
4
E
3
X)
ji
3
c
o
a.
°00
2
^
<
<D
O
C l.
1
0
-30
-20
-10
0
10
20
30
D C F ield (m T )
F ig. 4 .1 1 . S c h e m a tic o f o n e h a lf o f th e h y ster esis lo o p in the reg io n R+ w h ere the
arrrow s in d ica te th e d irectio n o f dc fie ld s w e e p . T h e d ash ed cu rv e rep resen ts the
V C and is sh o w n here o n both sid e s to e a s e co m p a r iso n . T h e p o in t to n ote is that
on fie ld reversal the h y ste r e sis lo o p retraces the V C , in d ica tin g that the lo o p is not
sy m m etric about AP mn.
154
0
0
H m ax = 17 m T
•
H m ax = 21 m T
■
H m ax = 10 m T
■
H
a
Q
'
V
♦d
«G
-5
X
£
-10
-15
-10
0
10
20
30
H0H (mT)
F ig. 4 .1 2 . T h e o b serv e d local field as a fu n ction o f the ap p lied field for three
d ifferen t m a x im a l fie ld ( |i Q
) v a lu e s for the c a se o f th e 10 |i.m Y B C O
p o w d er at 7 7 K and 10 G H z. N o te that - (i) the local field is a ffected by the
m a x im al fie ld and (ii) the local field reverses sign c lo s e to zero field .
155
5.0
D
Hmax = 20,7 mT
A
Hmax = 1 7 0 mT
O Hmax = 14.0 mT
□
□
*
□
□
o
0.0
u
° O O O
i i
D *
a
ss1
£
o
o
A
A
□
a
B
O
*0 A
°
*5.0 --------- -------------L...
0.0
-5.0
•
- 1
□
---- .--------- l
5.0
10.0
□
1
______
15.0
20
p H(mT)
F ig. 4 .1 3 .
T h e e stim a ted lo ca l fie ld as a fu n ctio n o f the ap p lied fie ld for three
d ifferen t m a x im a l fie ld (
v a lu e s for the c a se o f the 10 p m Y B C O
p o w d er at 5 0 K and 10 G H z.
156
.
In order to arrive at the V C , w e co n sid e red an a sse m b ly o f R SJs
ea c h w ith the sa m e r]n e x p o s e d to the sam e field H but h a v in g a d istrib u tion o f
e f fe c tiv e areas. A rg u a b ly , h y ster esis occu rs w h en the ap p lied field is not eq u al to
th e fie ld s e n se d by the R S Js,
S in c e o n e is d e a lin g w ith a gran u lar sa m p le , a
reasonab le assu m p tion to m ake is that the sam ple on ex p o su re to field w ill trap flux
at ran d om s ite s w h ic h are then so u r c e s o f
random fie ld s.
T o ob tain E q. 4 .2 ,
th erefore, w e b eg in w ith an e n se m b le o f R SJs each o b e y in g E q .3 .6 . but p la ced in
an e n v ir o n m e n t in w h ic h the a p p lied fie ld is a u g m en te d by random fie ld s , H r .
T h e applied field H is taken to be alon g the z d irection. E ach co m p o n en t o f H r is
a ssu m e d to h a v e a G a u ssia n d istribu tion
w ith
r.m .s. w id th s <7, = £7, = fT. = (7,
sin c e , in a p o w d er sa m p le there is n o p refered d irection . T o a cco u n t for S H, the
m o d e o f the z -c o m p o n e n t is set at a n o n -zero v a lu e, H m. T h e p o w e r ab sorp tion
can then be w ritten as
AP ( H )
2 e -tHm+ny/ o
2e
a
4no{Hm+l
T he d isc u ssio n is m ade sim p ler by con sid erin g the fo llo w in g points.
1) T h e h y s te r e s is lo o p can be c a lc u la te d by u s in g E q. ( 4 .3 ) fo r an y th ree
param eters H m,
and £7. It also turns out that instead o f g u e ss in g the v alu es for
the param eters o n e can read o f f their valu es by com p arin g the h y steresis lo o p to the
co rresp o n d in g V C . For e x a m p le , £7, is d eterm in ed by the fie ld v a lu e at w h ich the
a b so rp tio n A P on th e V C is eq u al to the m in im u m a b so rp tio n AP min o f the
157
h y s te r e s is lo o p .
In o th er w o r d s.
A Pnim = [ ^ ( ( 7 ) ] ^ .
T h e m in im u m in the
ab sorp tion ( AP min) is en tirely d eterm in ed by the rm s spread £7 o f the d istrib u tion
and s in c e
AP mm
in c r e a s e s w ith in c r e a sin g
m o n o to n ica ily w ith H max . W ith in crease in
in crea ses and this is in d icated in Fig. 4 .6 .
/ / nm ( s e e
AP ma as
F ig .
4 .5 ),
a
r is e s
in c r e a se s, AP IRA a lso
T h e param eter, H m, on the oth er hand
is not a co n sta n t but is co n tro lled by both the m axim al ap p lied fie ld as w e ll as the
instantaneous field and w ill be d isc u sse d shortly.
2 ) In th e r e g io n s R +
and R _ , co m p a riso n o f the h y ster esis lo o p to the V C w a s
u sed to g et H w c and o n e can u se Eq. 4 .2 to d escr ib e the h y ste r e sis lo o p , w h ere
the e ffe c tiv e field H eff = H ± H l o c . H o w e v e r , in th e reg io n R o,
4 .2
in stea d o f Eq.
it is better to u se Eq. 4 .3 w ith the param eters cr?K) and H m= 0 , s in c e , there is
n o sim p le c o n n e c tio n b etw e en
HLOC and
<7. It turns o u t that the r e g io n
R o is
a d eq u a tely d e scr ib ed by E q. 4 .3 w ith the v a lu e s H m= 0 and <7=0.1 H 0 to 0 .3 //„ .
A sa m p le c a lc u la tio n is sh o w n in F ig. 4 .1 4 .
T he ty p ica l v a lu e s o f a n ece ssa ry
su g g ests that in this region o n e o n ly requires sm all am ou n ts o f trapped flu x .
3) In the r e g io n s / f _ and R + , d H * 0 a n d is d ep en d en t on H m a.
addition to <7( H mja) w e n o w n eed a
* 0.
T h e r e fo r e , in
H o w e v e r , it turns ou t that in
eith er c a s e a s in g le v a lu e for H m is not ca p a b le o f d escr ib in g the h y ster esis loop .
T o d o this o n e w ill n eed to m ake H m d ep en d en t n ot o n ly on the m a x im u m field
/ / niax but a lso on the in stan tan eou s valu e o f field H . W e c o n sid e r th e R +
lo o p
first. S ettin g the left-h and sid e o f Eq. 4 .3 equal to the o b served v a lu e o f AP / a at a
fix e d H , the v a lu e o f H m w a s c a lc u la te d .
T h e fu ll lin e in F ig , 4 .1 0 is the
calcu lated H m( H ) for th e 10 p m Y B C O p o w d er sa m p le at 10 G H z and 7 7 K.
F ig s. 4 .1 5a and 4 . 1 5b sh o w s a sim ilar ca lcu la tio n for the sa m e sa m p le at 10 G H z,
7 7 K and 5 0 K r e s p e c tiv e ly for d ifferen t H mM v a lu e s.
158
A s can b e se e n H m( H )
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
F ig . 4 .1 4 .
C a lc u la te d lo o p s fo r th e r e g io n
u s in g E q. ( 4 .3 ) w ith th e fit
p a ra m eters (a ) cr = 0 ,3 m T , / / m = 0 , ^ 0/ / 0 - lm T and m a x im a l fie ld f i aH mn = 7
m T and (b ) o = 0.1 m T , H m = 0 , / i n/ / 0 = lm T and m axim al fie ld j i {)H miX = 3 .5
m T . N o te that in creasin g cr leads to increase in the irreversib le resiudal absorption
at zero field {
159
° M'q H
loc
60
H
' tI
o
50
'
40
S
3^
3a
X
= ? 20
10.0
12.5
15.0
F ig. 4 . 15a. T h e o b se r v e d lo c a l fie ld Pq H lqc and the c a lcu la te d Pq H m a s a
fu n ction o f the a p p lied field p H for m axim al fie ld o f p^
ca se o f the 10 p m Y B C O p o w d er sa m p le at 7 7 K and 10 G H z.
160
= 1 7 m T in the
40
M o*W X 10
T)
30
0«
OB
25
50
75
100
125
150
175
-10
N H ( X 1 0 4 T)
F ig . 4. 15b. T h e o b serv e d local field (Jq H loc and the c a lc u la te d pp H m as
a fu n ction o f the ap p lied fie ld p ()H fo r m a x im a l fie ld o f p 0 H max = 2. Im T
in the ca se o f the 10 p m Y B C O p o w d er sam p le at 5 0 K and 10 G H z.
c lo s e ly f o llo w s H LOC{ H ) and ca n th erefo re b e a c c e s s e d b y u s in g th e sim p le
co m p a riso n o f the V C and the loop.
4 ) In the reg io n R _ , 5 H and co n seq u en tly the H m v a lu e s are sm a ll, < 0 .3 H 0 . F ig.
4 .1 6 s h o w s the c a lc u la te d lo o p o b ta in e d u sin g Eq. 4 .3 . A g o o d a g re em e n t w ith
th e o b s e r v a tio n s is la c k in g fo r large fie ld s w h er e th e d ata s h o w s that the lo o p
o v erla p s th e V C . H o w e v e r , o n e d o es ob tain the o b serv e d
6 H< 0 .
A c o m p le te exp lan ation for such a co m p le x field d ep en d e n c e o f H m
is not at hand. H o w e v e r , the fo llo w in g scen a rio se e m s p la u sib le. C o n sid e rin g that
the sa m p les are granular in nature, e x p o s in g the sam p le to file d c o u ld result in flu x
trp p in g at ra n d o m site s.
It is a ls o a rg u a b le that th e trapp ed flu x p r o d u c e s a
d istrib u tion o f ran d om fie ld s at the ab sorb in g site s. T h e se ab sorb in g site s are the
ran d om su rfa c e d e fe c ts w h ic h act as w e a k lin k s or trapping s ite s and h a v e b een
m o d e lle d h ere as R S Js.
It ap p ears as if th e trapp ed flu x g r o w s large w h e n the
s w e e p d o w n is b eg u n but d u e to a d istrib u tion in trapping e n e r g ie s , so m e o f the
flu x 'leak s out' as H is reduced from H mm, th ereb y red u cin g the "bias" in H r and
letting H m —> 0 . Fu rth erm ore, by th e tim e H c r o s se s zero and is b ein g in crea sed
in the o p p o site d ir ectio n , o n ly w e a k lo ca l fie ld e ffe c ts rem ain and the d istrib u tion
s ta y s c e n te r e d at z e r o .
F ig . 4 .1 7 a is a p lo t o f H m/ H 0(O)) a s a fu n c tio n o f
H j H 0( 0 ) ) for the 10 (im Y B C O p o w d e r sa m p le at f i QH msa o f 2 2 m T at tw o
d iffe r e n t m ic r o w a v e fr e q u e n c ie s o f 10 G H z and 3 6 G H z.
T h e s c a lin g s u g g e sts
that, as b e fo r e , th e ju n c tio n s sa m p le the d c fie ld in u n its o f H n, th e fu n d am en tal
p aram eter o f L F IM A . F igu re 4 . 1 7b s h o w the sa m e s c a lin g fo r the 10 jam Y B C O
p o w d er at ^ 0/ / max o f
21 m T for 0 ) / 2 k o f 10 G H z and 3 6 G H z at 5 0 K
162
0.7
AP(H)
0.5
0.3
,
-
5.0
-
2.5
2.5
0
5.0
7.5
10.0
\xq H (mT) -►
F ig . 4 .1 6 . T h e c a lc u la te d h y ste r e sis lo o p o f reg io n /?_ob tain ed u s in g E q. (4 .3 )
w ith the fit param eters H 0 = 4 m T , o = 0 .4 m T and H m = 0 .2 m T . N o tic e that the
lo o p is r e v e r se d an d c o m p a r e s w e ll w ith the o b se r v e d R_ lo o p . T h e ca lc u la tio n
ap p roach es the V C tan gen tially as field is increased.
163
20
A
10 GHz
10
0
10
10
0
10
20
30
40
H /H
F ig . 4 .1 7 a .
T h e local fie ld as a fu n ction o f the ap p lied fie ld both sc a le d to the VC
field param eter fio H 0 at 10 G H z and 3 6 G H z for th e 10 ^ m Y B C O p o w d er at 77
ax= 2 2 m T . T h e sc a lin g su g g e sts that the R SJs s e n s e th e a p p lied
fie ld in u n its o f H0 .
164
■
4
36 GHz
1
0
■5
10
0
15
H/Hfl
F ig. 4 . 17b.
T h e lo ca l fie ld as a fu n ctio n o f the ap p lied field both sc a le d to the
V C field param eter ^
H q at 10 G H z and 3 6 G H z for the 10 p,m Y B C O p o w d er
at 5 0 K fo r jx 0 Hmaj= 21 m T . T h e sc a lin g su g g e sts that th e RSJ s se n se th e ap p lied
field in u n its o f H Q.
165
T h e m e ch a n ism su g g e ste d a b o v e to a cco u n t fo r the h y ste r e sis can
a ls o b e r e c o n c ile d w ith the n o n -h y s te r e tic m a g n e tiz a tio n o b s e r v e d in th ese
m a teria ls.
E ss e n tia lly the sm a ll current lo o p s can p rod u ce siz e a b le lo c a l fie ld s
w ith ou t altering the m a cro sco p ic m agn etic m om en t sig n ifica n tly . A rough estim ate
o f the kinds o f fie ld s resu ltin g from current lo o p s can b e m ad e as fo llo w s. Im agin e
a cylin drical current sh eet o f length X , the penetration depth, and o f radius r .
The
e ffe c tiv e d ip o le m o m en t f i eJf can then b e w ritten as pieff ~ x J cX r 3, J c b e in g the
cr itica l current d e n sity .
F or ty p ica l v a lu e s o f J t ~ 10lo A / m 2 , X ~ 1 0 “* m and
r - 10“* m the associated field
~ lm T . T h is is not to o far o f f from the
v a lu es o f HaH m required to d escrib e the h y steresis loop s. N ote that d ip o le m om en t
re su ltin g from su ch current lo o p s d o n ot m ak e s ig n ific a n t c o n tr ib u tio n to the
m agnetization.
H y s t e r e s i s in C r it i c a l C u r r e n t : It is w e ll k n o w n [9 0 . 9 1 , 9 2 ] that critical current in
granular H T S C o x id e sa m p les s h o w s h y ste r e sis as a fu n ctio n o f the a p p lied field .
O n ce a g ain , several m e c h a n ism s [3 4 , 9 3 ] h ave b een p rop osed . S o m e are b ased on
the critical state m o d el w h ile so m e oth ers [9 4 ] u se J o sep h son ju n ctio n arrays. T he
c o m p le x ity o f the p h en o m en o n is su ch that any m icr o sco p ic picture w o u ld h ave to
u se a n um ber o f p aram eters. A lte rn a tiv ely , w e p ro p o se in the fo llo w in g a sim p le
in terp retation o f the R SJ eq u a tio n s p rese n ted in C h ap ter III and sh o w that it can
a ccou n t for the h y steresis in 7 ( / / ) .
In term s o f the critical current, Eq. (3 .2 5 ) can be form ally w ritten as
w h ere >3is sa m p le d ep en d en t and in c lu d es all field -in d e p e n d en t param eters. From
E q. 4 .4 o n e e x p e c ts (i) the m o n o to n ic in c r e a se in m a g n e to a b s o r p tio n w ith
in crea sin g H (V C ) to co r resp o n d to a m o n o to n ic d e c r e a se in
m a g n e to a b s o r p tio n to c o r r e sp o n d to a h y s te r e s is in
7,
7 ,. (ii) h y ster etic
i .e .,
th e
lo s s
in
m a g n eto a b so rp tio n on fie ld reduction to co rresp o n d to an in crea se in 7, w h ile the
m in im um in A P ( H ) co r resp o n d s to a m a x im u m in
A sc h e m a tic o f the 7
h y s te r e s is lo o p is sh o w n in F ig. 4 .1 8 . (iii) C arryin g th is a n a lo g y fu rth er, the
ch aracteristics o f the J t h y steresis lo o p sh ou ld d ep en d on the m ax im u m o f the field
s w e e p , i.e .,
fu n ctio n o f
In particular, the m a x im a o f the 7, ( / / ) lo o p s sh o u ld be a
and a rev ersa l in the lo o p structure (R +
and R _ r e g im e s )
sh o u ld a lso be o b serv e d .
T h e a b o v e e x p la n a tio n o ffers the in trigu in g p o ss ib lity o f o b ta in in g
the fie ld -d e p e n d e n t current lo o p s w ith o u t h a v in g to attach lead s to the sa m p le as
w e ll as u sin g very sm all ( ~ 1 0 7 A / m : ) current d e n sitie s. A s n o ted a b o v e , several
a u th ors h a v e o b se r v e d h y ster etic cu rren t-field relation s in granular sa m p le s u sin g
c o n v e n tio n a l tran sp ort m e a su r e m e n ts.
H o w e v e r , u ntil r e c e n tly , there w a s n o
report o f a ch an ge in the 7 v s. H lo o p ch arateristics w ith / / mas su ch as d is c u ss e d
a b o v e . R e c e n tly , M ah el e t a l. [6 2 ] have o b serv e d su ch a s w itc h -o v e r in lo w -fie ld
J C( H )
m e a s u r e m e n ts
in g ra n u la r Y B C O
film s as a fu n c tio n o f
f.iuH max.
I n c id e n tly , J a c k ie w ic z e t a l. [9 5 ] s h o w that a stra ig h tfo rw a rd u s e o f th e B ea n
m o d e l to a cc o u n t fo r th e h y ste r e sis in critica l current in gran u lar H T S C sa m p les
d o e s n ot rep ro d u ce all th e o b se r v e d a n o m a lie s.
In stead , th ey p r o p o se a m o d e l,
b a sed on w ea k lin k s e x p o s e d to an e ffe c tiv e m a g n etic fie ld w h ich resu lts from the
d ip o la r fie ld d u e to trapp ed flu x ad d in g v e c to r ia lly to the a p p lie d fie ld .
167
T h eir
c a lc u la tio n n ot o n ly p red icts that the lo o p structure is in flu e n c e d by th e m a x im a l
fie ld but a lso y ie ld s the critical current lo o p structure w h ich co rrep o n d s, b a sed on
ou r interpretation, to the m icro w a v e absorption in the region 7^,.
168
0
H — ►
F ig . 4 .1 8
S c h e m a tic rep resen tation o f the critical cuu ren t d en sity Jc ( H ).
N o tic e
that J ( 0 ) > 7 ' ( 0 ) . T h e arrow s represent the d irection o f the m a g n etic fie ld ch an ge
(c f. F ig . 1, R e f. 9 2 ).
169
h
9
i n
&
Chapter V
Summary
In su m m m a ry , a sy te m a tic ex p er im en ta l stu d y o f th e lo w fie ld
m a gn etoab sorp tion , in granualar cuprates has been con d u cted . T he o b serv a tio n s o f
m icr o w a v e m agn eto a b so rp tio n w ere carried ou t o v er a w id e range (2 G H z to 6 0
G H z) o f fr e q u e n c ie s and tem p eratu res.
O ur m e a su r em en ts o f
b oth
the static
r e sp o n se and the r e sp o n se in a m ic r o w a v e fie ld on the sa m e p o w d e r H T S C
sa m p le s sh o w that there is n o sim p le c o n n e c tio n b e tw e e n the tw o p h e n o m en a .
T h is h e lp e d u s to c o n c lu d e that any m o d el that a s s o c ia te s th e fie ld in d u ced
m ic r o w a v e lo s s to an e f f e c t iv e B w ith in the sa m p le w a s in ap p rop riate.
We
p ro p o se, instead , that the sam p le be treated as a c o lle c tio n o f r e sis tiv e ly shunted
J o se p h so n ju n c tio n s .
In th is m o d el the lo s s in m ic r o w a v e p o w e r d u e to the
extern ally applied dc field is the direct co n seq u en ce o f the field ind u ced dam p in g o f
the J o se p h so n
current.
A s w e h ave sh o w n , su ch a s im p le R S J m o d el y ie ld s
ex cellen t agreem ent w ith the data.
In addition to a co m p le te d escrip tion o f the virgin cu rv e, w e have
c o n d u c te d a d e ta ile d stu d y o f the fie ld and tem p eratu re d e p e n d e n c e s o f the
h y ster etic m a g n eto a b so rp tio n .
For rea so n s c ite d in C h ap ter IV , a m ic r o sc o p ic
m o d el for the p h en om en on is a form id ab le task. H o w ev e r, w e sh o w that a sim p le
e x te n s io n o f th e RSJ m o d el a llo w s for a re a so n a b ly g o o d d e s c r ip tio n o f the
h y steretic resp on se.
170
L a stly , it is to b e a c k n o w le d g e d that g iv e n th e c o m p le x it ie s o f
granular H T S C m aterial, it is rem ark ab le that su ch a sim p le m o d e l is c a p a b le o f
rep rod u cin g m o st o f the ob serv a tio n s..
171
References
1.
J. G . B ed n o rz and K. A. M iiller, Z. P h ys B 6 4 . 189 (1 9 8 6 )
2.
A . M . Portis, E a r li e r a n d R e c e n t A s p e c t s o f S u p e r c o n d u c tiv ity , S prin ger
series in S o lid S tate S c ie n c e , ed s. J. G . B ed n o r z and K. A . M u ller (S p ringer,
B erlin , 1990).
3.
K. K hachuturyan, E. R. W eb er, P. T ejed er, A . M . S ta cy and A . M . Portis,
P h ys. R e v . B 3 6 , 8 3 0 9 (1 9 8 7 ).
4.
K .W . B la z e y , A . M . Portis and J. G . B ed n o rz, S o lid S tate C o m m . 6 5 , 1153,
1988.
5.
S . V . B hatt, P. G a n g u ly , P. V . R am akrishnan and C. N . R. R ao, J. P h ys.
C 2 0 , L 5 5 9 (1 9 8 7 ).
6.
E. J. P a k u lis and T . O sada, P h ys. R ev . B 3 7 , 5 9 4 0 ( 1 9 8 8 )
7.
A . G o u ld , S. M . B h agat, M . A . M an h eim er, J. A p p l. P h y s. 6 7 , 5 0 2 0
(1 9 9 0 ) .
8.
M . M a h el, R. H lu b in a and S. B en a ck a , P h y sic a C 1 6 9 . 4 2 9 (1 9 9 0 ).
9.
A . M . P ortis, D . W . C o o k e , E. R. G ray, P. N . A ren dt, C . L. B o h n , J. R.
D e la y e n , C. T . R o c h e , M . H ein , N . K lein , G . M u ller, S . O rbach and H. P iei,
A p p l. P h ys. L ett. 5 8 , 3 0 7 (1 9 9 1 ).
10.
T . L. H y lto n , A . K apituln ik , M . R. B e a s le y , J. P. C arin i, L. D rab eck and G.
G rtiner, A p p l. P h ys. L ett. 5 3 , 1343 (1 9 8 8 ).
11.
R. F astam p a, M . G iura, R. M arcon and C. M atacotta, E u rop h ys. L ett. 6 , 2 6 5
(1 9 8 8 ) .
12.
M . X . H u an g, S. M . B h agat, A . T . F in d ik o g lu and T . V e n k a tesa n , J. A p p l.
P h y s. 6 8 , 6 3 1 5 (1 9 9 0 ).
13.
J. H albritter, Journal o f A llo y s and C o m p o u n d s
172
1 9 5 . 5 7 9 (1 9 9 3 ).
14.
A . A . Z h u k o v , D. A . K om ark ov. G. K arapetrov. S. N. G o rd eev and R. I.
A n to n o v , S u p ercon d . S ci. T ech n o l. 5, 3 3 8 (1 9 9 2 ).
15.
J. H albritter, P r o c . 5th S u p e r c o n d u c tio n R F W o r k s h o p (H am b u rg, 1991).
16.
A . D u lc ic , B . R ak vin and M . P o zek , E u roph ys. L ett. 1 0 , 5 9 3 (1 9 8 9 ).
17.
J. S . R am achan d ran , M . X . H u an g, S. M . B h a g a t, K. K ish and S. T y a g i,
P h y sic a C 2 0 2 , 151 (1 9 9 2 ).
18.
J. S. R am achan d ran , S. E. L o fla n d , S. M . B h agat, M . A . M an h eim er and S.
T y a g i, S o lid State C o m m . 9 3 , 671 (1 9 9 5 ).
19.
J. S. R am achan d ran , M . X . H uan g and S. M . B h agat, P h y sic a C 2 3 4 . 173
(1 9 9 4 ) .
20.
J. S. R am achandran and S. M . B hagat, IE EE Trans, o f A p p lied
S u p er co n d u ctiv ity , In press.
21.
W . M e issn e r and R. O c h se n fe ld , N a tu rw isse n sc h a fter 2 1 , 7 8 7 (1 9 3 3 ).
22.
F. L on d on and H . L o n d o n , P h y sic a C 2 , 341 (1 9 3 5 ).
23.
M . T ink h am , In tr o d u c tio n to S u p e r c o n d u c tiv ity (M cG ra w -H ill, N e w Y ork,
1 9 7 5 ).
24.
T. P. O rlan do and K . A . D e lin , F o u n d a tio n s o f A p p l i e d S u p e r c o n d u c tiv ity
(A d d is o n -W e s le y , R e d d in g , M A , 1991).
25.
C. J. Jou, E. R. W eb er, J. W ashburn and W . A . S o ffa , A p p l. P h ys. L ett. 5 2 ,
3 2 6 (1 9 8 8 ), A . O u rm azd , J. A . R en tsch ler, W . J. S k o c p o l and D . W , Joh n son ,
Jr., P h ys. R ev . B 3 6 , 3 2 6 (1 9 8 8 ).
26.
P. G . D e G e n n e s, S u p e r c o n d u c ti v it y o f M e t a l s a n d A l l o y s (B en ja m in , N e w
Y o rk , 1 9 6 6 ).
27.
C . P. B ea n , P h ys. R e v . L ett. 8 , 2 5 0 (1 9 6 2 ).
28.
M . C yrot and D . Pavuna, I n tro d u ctio n to S u p e r c o n d u c tiv ity a n d H igh-T,
M a te r ia ls (W orld S c ie n tific , S in gap ore, 1992)
173
29.
V . A m b eg a o k a r and A. B aratoff, P h ys. R ev. Lett. 1 0 , 4 8 6 (1 9 6 3 ).
30.
A . B aron e and G. Paterno, P h y s i c s a n d A p p l ic a ti o n s o f the J o s e p h s o n Effect
W ile y , N e w Y ork , 1982).
31.
R. L. P eterson and J. W . E kin, P hys. R ev. B 3 2 , 9 8 4 8 (1 9 8 8 ).
32.
K. -H . M u ller and D . N . M a tth ew s, P h y sica C 2 0 6 . 2 9 5 (1 9 9 3 ).
33.
R. M arcon , R . F astam pa, M . G iura and C . M atacotta, P h ys. R ev . B 3 9 .
2 7 9 6 (1 9 8 9 ).
34.
J. D . F ranson and J. E. M ercereau , J. A p p l. P h ys. 4 7 , 3261 (1 9 7 6 ).
3 5.
A . M . Portis, E le c tr o d y n a m ic s o f H ig h - T e m p e r a tu r e S u p e r c o n d u c to r s ,
L ecture N o te s in P h y sic s - V o l. 4 8 (W orld S c ie n tific , S in g a p o re, 1992).
36.
M . X . H u an g, P h .D D isserta tio n , U n iv ersity o f M arylan d , C o lle g e Park,
M arylan d , u n p u b lish ed (1 9 9 3 ).
37.
N . D . S p e n c er and A . L aw ren ce R o e , A C S S y m p o siu m S eries, P h y sica l
C h em istry o f H igh-T em p eratu re S u p ercon d u ctors 3 7 7 . C hap. X II (1 9 8 8 ).
38.
N . D . S p e n c er, S . D . M urphy, G . S h a w , A . G o u ld , E. M . Jack son and S . M.
B h agat, Jpn. J. A p p l. P h ys. 2 8 , 1 5 6 4 (1 9 8 9 ).
39.
G . S h a w , P h .D D isserta tio n , U n iv ersity o f M arylan d , C o lle g e Park, M aryland,
u n p u b lish ed (1 9 9 1 ).
40.
A . N ath , Z. H a m o n n a y , S. D . T y a g i, Y . W e i, G . W . Jang and C . C . C han,
P h y sic a C J 7 1 , 4 0 6 (1 9 9 0 ).
41.
K. K ish , S . T y a g i and C. K rafft, P h ys. R e v . B 4 4 . 2 2 5 (1 9 9 1 ).
42.
S. M . B h agat, D . J. W eb b and M . A . M an h eim er, Jour. M a g . and M ag. M at.
53 , 2 0 9 (1 9 8 5 ).
43.
A . G o u ld , M . X . H uan g, S. M . B h agat and S. T y a g i, J. A p p l. P h ys. 6 9 ,
4 8 8 0 (1 9 9 1 ).
174
44.
R. D u rn y , J. H antala, S . D u ch arm e, B . L ee, O. G . S y m k o , P. C. T aylor, D.
J. Z h e n g and J. A . X u , P h ys, R ev. B j36, 2361 (1 9 8 7 ).
45.
K. W . B la z e y , K. A . M u ller, J. G . B ed n o rz, W . B erlin ger, G . A m oretti, E.
B u lu g g iu , A . V era and F. C. M atacotta, P h ys. R ev. B 3 6 , 7241 (1 9 8 7 ).
46.
V . F o u k is, O . D ob b ert, K. P. D in s e , M . L eh in g , T. W o lf and W . G old ack er,
P h y sic a C 1 5 6 . 4 6 7 (1 9 8 8 ).
47.
A . M . P ortis, K . W . B la z e y , K. A. M u ller and J. G. B ed n o rz, E u rop h ys. Lett. 5,
4 6 7 (1 9 8 8 ).
48.
K . W . B la z e y , A . M . Portis and J. G . B ed n o rz, S o lid State C o m m . 7 0 , 7 2 9
(1 9 8 9 ) .
49.
G . S h a w and S . M . B h a g a t, J. A p p l. P h ys. 6 2 , 5 3 7 3 (1 9 9 1 ).
50.
G . S h a w , S. M . B h agat, N . D . S p en cer, J. E. C row and S . T y a g i, P h y sic a C
1 6 9 . 2 5 7 (1 9 9 0 ).
51.
E. M . J a ck so n , S. B . L ia o , J, S ilv is, A . H. S w ih art, S. M . B h agat, R.
C ritten d en , R. E. G lo v e r III and M . A . M an h eim er, P h y sic a C 1 5 2 . 125
(1 9 8 8 ).
52.
A . G o u ld , E. M . J ack son , K. R enou ard , R. C ritten d en , S. M . B h agat, N . D .
S p e n c er, L. E. D o lh ert and R. F. W orm sb ech er, P h y sic a C 1 5 6 . 5 5 5 (1 9 8 8 ).
53.
F. L o n d o n , S u p e r f l u i d s , V o l 1 (D o v e r , N e w Y ork, 1961).
54.
A . G o u ld , S. D . T y a g i and S. M . B h agat, IE EE Trans, on M ag. 2 5 , 3 2 2 4
(1 9 8 9 ) .
55.
G . S h a w , R. C ritterden, Y . M . M oon and S. M . B h agat, J. M agn . M agn .
M aterials 7 8 , L I 3 (1 9 8 9 ).
56.
J. R. C le m and M . W . C o ffe y , P h ys. R e v . B 4 2 , 6 2 0 9 (1 9 9 0 ).
57.
B . M u h ls c h le g e l, Z . P h ys. 1 5 5 . 3 1 3 (1 9 5 9 ).
58.
R. A . F errell, P h y sic a C 1 7 5 . 1 (1 9 8 9 ).
175
59.
M . A. M an h eim er, S. L oflan d and A. G ou ld , P h y sic a C 183. 3 2 4 (1 9 9 1 )
60.
M . G iura, R. M acron , R. Fastam pa and E. S ilv a . P h ys. R ev. B 4 5 , 7 3 8 7
(1 9 9 2 ).
61.
A . D u lc ic , R. H. C repeau, J. H. Freed, L. F. S c h n e e m e y e r and J. V.
W a sz c z a k , P h ys. R ev. B 4 2 , 2 1 5 5 (1 9 9 0 ).
62.
M . M ah el, M . D arula and S. B en ack a, S up ercon d. S ci. T ech n o l. 6 , 1 1 2
1 9 9 3 ).
63.
B . C z y z a k , J. S ta n k o w sk i and J. M artinek, P h y sic a C 2 0 1 . 3 7 9 (1 9 9 2 ).
64.
S . T y a g i, P rivate C om m u n ication .
65.
V . S esh u B a i, P. V . Patanjali, S. M . B h agat and S. T y a g i, Journal o f
S u p er co n d u ctiv ity , In Press; V . S esh u B ai, P. V . Patanjali and S. M .
B h a g a t, S o lid State C o m m . 9 0 , 8 0 9 (1 9 9 4 ).
66.
M . X . H u an g, S . M . B h a g a t, A . T. F in d ik o g lu , T . V en k a tesa n , M . A.
M an h eim er and S. T y a g i, P h v sica C 193. 421 (1 9 9 2 ).
67.
A . G o u ld , S. M . B h agat, F. C. W e llsto o d and S. T y a g i, S o lid S tate C o m m . 8 1 ,
3 3 9 (1 9 9 2 ).
68.
R. N avarro and L. J. C am p b ell, P h ys. R ev. B 4 4 , 1 0 1 4 6 (1 9 9 1 ).
69.
J. R. C lem , P h y sic a C 1 5 3 -1 5 5 . 5 0 (1 9 8 8 ).
70.
A . M . S tew art, S u p ercon d . S c i. T ech n o l. 4 , 3 2 (1 9 9 1 ),
71.
L. Ji, M . S. R z c h o w sk i, A . A nand and M . T ink h am , P h ys. R ev . B 4 7 , 4 7 0
(1 9 9 3 ) .
72.
L . Ji, M . S . R z c h o w sk i and M . T ink h am , P h ys. R ev. B 4 2 , 4 8 3 8 (1 9 9 0 ).
73.
S e e fo r e x a m p le , J. R. C lem and V . G. K ogan , Jpn. J. A p p l. P h y s. 2 6 ,
1161 (1987).
176
7 4.
A . M . Portis, E le c tr o d y n a m ic s o f H ig h -T e m p e r a tu r e S u p e r c o n d u c to r s ,
C hapters 11 and 12 (L ecture N o te s in P h y sic s - V o l. 4 8 , W orld S c ie n tific ,
S in g a p o re, 1992).
75.
F. W e llsto o d , P rivate C om m u n ication .
76.
R. F astam p a, M . G iura, R. M arcon and C. M atacotta, E uroph ys. L ett. 6 , 265
(1 9 8 8 ) .
77.
E. S ilv a , M . G iura, R . M arcon , R. F astam p a, G . B a lestrin o , M . M arin elli and E.
M ila n i, P h ys. R ev . B 4 5 , 1 2 5 6 6 (1 9 9 2 ).
78.
J. M anhart, P. C h audhari, D . D im o s, C. C . T su ei and T. R . M cG u ire, P hys.
R e v . L ett. 6 L 2 4 7 6 (1 9 8 8 ).
79.
S . S . L aderm an , R. C . T aber, R. D . J a co w itz, J. L. M ott, C . B . E o m , T . L.
H y lto n , A . F. M arsh all, T. H. G eb a lle and M . R. B e a s le y , P h y s. R ev . B 4 3 ,
2 9 2 2 (1 9 9 1 ).
80.
S. R. R em illa rd , M . E. R e e v e s and F. J. R achford, J. A p p l. P h ys. 7 5 , 4 1 0 3
(1 9 9 4 ) .
81.
A . F. K o s h e le v , I. L e v ie v and R. S. P ap ik yam , JE T P 7 6 , 4 5 9 (1 9 9 3 ).
82.
J. W o sik , L, M . X ie , R. C hau, A . S am aan , J. C. W o lfe , V . S elv a m a n ick a m
and K. S alam a, P h ys. R ev. B 4 7 , 8 9 6 8 (1 9 9 3 ).
83.
M . M a h el, IE E E Trans, on A p p lied S u p ercon d u ctivity 3, 1435 (1 9 9 3 ).
84.
A . N ish id a and K. H orai, S o lid S tate C om m . 7 4 , 9 4 7 (1 9 9 0 ).
85.
S . L o fla n d , Private C om m u n ication .
86.
M . M a h el, R . A d am , M . D arula, S . C hrom ik and S. B en ack a, P h y sic a C 2 0 2 .
2 4 3 (1 9 9 2 ).
87.
S e e for e x a m p le A. U m e z a w a , G . W . C rabtree, K. G , V an d ervoort, U .
W e lp , W . K. K w o k and J. Z. L iu, P h y sic a C 1 6 2 -1 6 4 . 7 3 3 (1 9 8 9 ).
88.
T . K o y a m a , N . T a k e za w a and M . T ach ik i, P h y sic a C 1 6 8 . 6 9 (1 9 9 0 ).
177
89.
J. A n n ett, N . G o ld e n fe ld and S. R. R ean, P h ys. R ev. B 4 3 , 2 7 7 8 (1 9 9 1 ).
90.
C. P. B ean and J. D . L iv in g sto n , P h ys. R ev. L ett 1 2 , 14 (1 9 6 4 ).
91.
J. E. E vetts and B . A . G lo w a ck i, C r y o g en ic s 2 8 , 641 (1 9 8 8 ).
92.
M .J E. M c H e n r y , M . P. M a ley and J. O. W illis, P h ys. R ev. B 4 0 , 2 6 6 6
(1 9 8 9 ) .
93.
T. R. A s k e w , R. B . F lip p en , K. J. L eary and M . N . K unchur, J. M ater. R es. 6,
1 (1 9 9 1 ).
94.
H . D e rsch and G. B latter, P h ys. R ev. B 3 8 , 11391 (1 9 8 8 ).
95.
R. N avarro and L. J. C a m p b ell, S u p ercon d . S c i and T ech n o l. 5, S I 21
(1 9 9 2 ) .
96.
J. J a c k ie w ic z and J. L e sz c z y n sk i, P ro ceed in g s o f the 6th International
W o rk sh o p on C ritical C urrents, ed ited by J. E. E vetts, C am b rid ge, U K ,
1 9 9 1 ), p. 5 3 8 7 .
178
Документ
Категория
Без категории
Просмотров
0
Размер файла
4 313 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа