Revista Brasileira de Física, Vol. 8, N? 1, 1978
Magnetic Hysteresis and Flux Prof iles in Type- I Superconductors
GUNTER ZERWECK
Instituto de Física "Gleb Wataghin", Universidade Estadual de Campinas*, Campinas SP
Recebido em 23 de Novembro de 1977
Dedicated to the memory of Dr. Herman Trauble
The c r i t i c a l
interaction
(" pinning" )
current
of
of
type- l
superconductors
quantized magnetic
. Measurements o f
f l u x lines
i s determined by the
with
inhomogeneities
the p r o f i l e s o f the magnet i c i n d u c t i o n are
e q u i v a l e n t , i n many r e s p e c t s , t o c r i t i c a l
c u r r e n t rneasurements, and f r e -
q u e n t l y are much simpler. A f t e r a s h o r t i n t r o d u c t i o n about the general
p r o p e r t i e s o f type-l
superconductors,
this article
microscopic p i n n i n g mechanisms, t h e i r s t a t i s t i c
croscopic volume p i n n i n g f o r c e ,
del i n c l u d i n g surface e f f e c t s .
reviews t h e
main
cooperation t o the ma-
and the general i z e d c r i t i c a l s t a t e moThe b a s i c methods o f measurement
are
described, together w i t h some important r e s u l t s about b u l k , surface, and
near- surface p i n n i n g . The f i n a l discussion stresses e s p e c i a l l y two
ob-
s e r v a t i o n s s t i I l h a r d l y understood: the d i f f e r e n t e f f e c t s o f the s u r f a ce and the i n f l u e n c e o f magnetic f i e l d h i s t o r y .
A c o r r e n t e c r i t i c a de supercondutores de
tipo-l
é determinada
i nteração de L
i nhas quant i zadas de f l u x o magnético com
("pi nni ng", aprisionamento)
. Medi das
pela
i nomogene i dades
de p e r f i s da i ndução magnét i ca são
equivalentes, em muitos aspectos, a medidas da c o r r e n t e c r i t i c a , e f r e quentemente bem mais simples.
Após uma introdução c u r t a sobre as
priedades g e r a i s de supercondutores de t i p o - l l , e s t e a r t i g o
uma resenha dos mecanismos microscópicos de aprisionamento mais
tantes, a cooperação e s t a t í s t i c a deles
Posta! Address: C.P.
1170, 13.100
-
impor-
à f o r ç a macroscópica por volume,
e o modelo do estado c r i t i c o generalizado,
*
pro-
apresenta
i n c l u i n d o e f e i t o s da
Campinas SP
super-
f ície.
Descrevem-se os métodos básicos da medida, e alguns
r e s u l tados
importantes sobre aprisionamento no volume, na s u p e r f í c i e e nas
midades da s u p e r f í c i e .
proxi-
A discussão f i n a l acentua especialmente duas ob-
servações ainda pou :o entendi das: os d i f e r e n t e s e f e i t o s de s u p e r f í c i e e
a i n f l u ê n c i a da h i s t ó r i a do campo magnético.
INTRODUCTION
A p p l i c a t i o n o f s u p e r c o n d u c t i v i t y began i n 1961 when Kunzler e t aZ. d i s coveredl
t h a t the compound Nb,Sn
destroying superconductivity.
i s able t o c a r r y a h i gh c u r r e n t without
Since then, h i g h l y s o p h i s t i c a t e d conduc-
t o r s have been developed, mainly f o r the p r o d u c t i o n o f magnet c o i l s f o r
a wi de range o f appl i cations; rangi ng f rom the l a b o r a t o r y magnets for bas i c research purpose t o the l e v i t a t i n g magnets i n high-speed t r a i n systems6'".
Another f i e l d where s u p e r c o n d u c t i v i t y rnay become o f increasing
p r a c t i c a l i n t e r e s t i s the transmission o f e l e c t r i c power
i n cryogenic
cables, a p r o j e c t which i s p r e s e n t l y being s t u d i e d i n a number o f labor a t o r i e s a11 over t h e w o r l d (see, e.g.,
~ e f . 1 5 ) . A survey o f the a c t u a l
s t a t e o f appl i c a t i o n o f s u p e r c o n d u c t i v i t y was pub 1 i s h e d
r e c e n t l y by
Schwartz and ~ o n e r ~ .
AI1 these devices make use o f t h e a b i l i t y o f t y p e - l l superconductors t o
This.
cric a r r y a l o s s l e s s c u r r e n t up t o a c e r t a i n c r i t i c a l c u r r e n t Ic
t i c a l c u r r e n t i s a f u n c t i o n o f temperature and magnetic f i e l d ,
and
is
mainly determined by t h e i n t e r a c t i o n o f magnetic f l u x v o r t i c e s w i t h extended c r y s t a l imperfect:ions l i k e p r e c i p i t a t e s , d i s l o c a t i o n
g r a i n boundaries.
walls,
or
A t r a n s p o r t c u r r e n t creates a f o r c e on the v o r t i c e s ,
and t o keep them from moving, which would cause power d i s s i p a t i o n , they
must be "pinned".
The s t r o n g e r the i n t e r a c t i o n between defects and f l u x
v o r t i c e s , the h i g h e r the c r i t i c a 1 c u r r e n t .
On the o t h e r hand, the c r i -
t i c a l c u r r e n t o f an " i d e a l " type-l l superconductor ( s i n g l e c r y s t a l
thout any defects)
i n the mixed s t a t e would be zero, even i n
wi-
materiais
w i t h h i g h c r i t i c a 1 temperature and h i g h c r i t i c a 1 f i e l d s .
The sameinteraction prevents, i n magneti z a t i o n e ~ e r i m e n t s ,the f l u x v o r t i
-
ces from reaching t h e i r thermdynamic equi 1 i b r i u m d i s t r i b u t i o n , thus a l -
lowing f o r the presence o f f l u x gradients w i t h i n a b u l k superconductor,
and g i v i n g r i s e t o a h y s t e r e t i c magnetization curve.
This equivalence between a t r a n s p o r t c u r r e n t and a f l u x densi t y gradient,
i n t y p e - l l superconductors,
i s t h e basis f o r the experimental study o f
.
"pinni ng" forces i n m d e l systems by magneti z a t ion measurements Such model
systems, each o f which i d e a l l y represents o n l y a s i n g l e i n t e r a c t i o n mechanism, i n s t e a d o f the m e t a l l u r g i c a l l y complicated s t a t e o f the
tech-
n i c a l superconductor, are, e.g.,
type:
dislocations,
s i n g l e c r y s t a l s wi t h one d e f e c t
i r r a d i a t i o n defects, o r p r e c i p i t a t e s . Because o f the h i g h
c r i t i c a 1 c u r r e n t s (up t o 106~/cm2), sample p r e p a r a t i o n f o r magnetization measurements (long c y l i n d r i c a l rods) i s much e a s i e r i n such
systems
than f o r d i r e c t current-v01 tage measurements ( t h i n tapes o r threads).The
purpose o f t h e present a r t i c l e i s t o describe
how much
information
about p i n n i n g forces can be drawn from magnetization measurements,
p e c i a l l y from the so c a l l e d "minor h y s t e r e s i s loops".
t o present a complete review o f i r r e v e r s i b l e p r o p e r t i e s o f t y p e - l I
perconductors.
es-
intend
We do n o t
su-
Thi s was excel l e n t l y done by Campbell and ~ v e t t s ' and,
more recently, by u l l m a i e r 9 .
hensive b i b l i o g r a p h y .
Both reviews a l s o c o n t a i n a more
compre-
D e t a i l e d i n f o r m a t i o n about many s p e c i f i c problems
may be obtained from the proceedings o f the discussion meeting
Andreasberg i n 1974, Ref . l O .
Melvi 1l e ' s review paperll
g r e a t influence of t h e surface on a.C.
in St.
s t resses
losses, which are c l o s e l y
ted t o the magneti z a t i o n h y s t e r e s i s t r e a t e d i n the present paper.
a r t i c l e o f L i v i n g s t o n and schadler12, although 13 years o l d , may
the
relaThe
still
serve as a valuable i n t r o d u c t i o n t o t h e whole f i e l d o f r e l a t i o n s between
superconductivi t y and metal l u r g y .
I n Section 1, a s h o r t i n t r o d u c t i o n t o t h e general p r o p e r t i e s o f type-l l
superconductors i s presented. The second Section descri bes, i n a qual i
-
t a t i v e r a t h e r than i n a mathematically exact way, the main
microscopic
i n t e r a c t i o n mechanisms and the s t a t i s t i c summation o f these
elementary
forces t o the macroscopic volume p i n n i n g force.
One o f the m s t f r u i t -
f u l ideas f o r the understanding o f i r r e v e r s i b l e e f f e c t s
perconductors was t h e concept o f
Bean16,
the c r i t i c a l s t a t e ,
i n t y p e - l l su-
i ntroduced
which w i l l be t r e a t e d i n Section 3, followed by
a
by
review o f
d i f f e r e n t methods o f measurement. The l a s t two Sections present some imp o r t a n t experimental r e s u l t s about bulk and surface pinning,and discuss
r
s p e c i a l l y the d i f f e r e n t e f f e c t s o f surface
and near- surface p i n n i n g as
w e l l as the i n f l u e n c e o f magnetic f i e l d h i s t o r y .
1. TYPE-II SUPERCONDUCTORS
This S e c t i o n i s j u s t t o g i v e a concise i n t r o d u c t i o n t o the
equilibrium
p r o p e r t i e s of t y p e - l l superconductors and the b a s i c phenomeno lo g i c a I
concepts f o r d e s c r i b i n g them. For more d e t a i 1 s,
I must r e f e r t o one of the
textbooks 1 i s t e d i n the f i r s t p a r t o f the b i b ~ i o ~ r a ~ h ~ l - ~ .
The d i f f e r e n c e between type- l and t y p e - l l superconductors i s b e s t i 1 l u s t r a t e d by t h e i r e q u i l i b r i u m r e l a t i o n betweenmagnetic i n d u c t i o n
d e n s i t y ) B and the a p p l i e d f i e l d H (Figs. l a and l b ) .
(flux
Type-l supercon-
ductors are c h a r a t e r i z e d by a complete e x c l u s i o n of the m a g n e t i c f l u x
( p e r f e c t diamagnetism) up t o the c r i t i c a 1 f i e l d Hc,
tion
t o the normal s t a t e takes place.
where
the transi-
I n t y p e - l l superconductors, the
surface energy between superconducting and normal
phases
i s negative,
thus t h e f r e e energy can be decreased, i n f i e l d s above the lower c r i t i c a l f i e l d Hcl,
by a s p l i t t i n g i n t o normal and superconducting "domins"
on a very f i n e scale.
magnetic f l u x ,
ween Hcl
This i s accomplished by a p a r t i a 1 p e n e t r a t i o n o f
r e f l e c t e d by t h a t p a r t o f the B-H curve o f F i g . l b
and Hc2 ("mixed s t a t e " ) . At the upper c r i t i c a 1 f i e l d Hc2,
f l u x p e n e t r a t i o n i s complete and the b u l k o f the specimen becomes
betthe
nor-
mal.
For p r a c t i c a l reasotis ( h i g h e r r e s o l u t i o n o f the o r d i n a t e ) , the m g n e t i z a t i o n curve - M ( H )
i s glven i n s t e a d o f the i n d u c t i o n curve B(H) i n most
cases ( ~ i g . 1 ~ ) .The magnetization M may be defined by
a1though i t does n o t have the d i r e c t physi c a l meani ng o f a magneti c d i pole d e n s i t y as, e.g.,
i n ferromagnets.
With t h i s d e f i n i t i o n , the i n i -
t i a 1 slope o f the magnet:ization curve i n the p e r f e c t l y diamagnetic s t a t e becomes -1.
A t h e o r e t i c a l descr , i p t i o n o f t h i s behavior i s provided by the Ginzburg-Landau equations17, twa coupled n o n l i n e a r d i f f e r e n t i a l equations f o r the
228
Fig.1
-
Equilibrium induction vs. f i e l d , and magnetization curves o f su-
perconductors: a ) type- l superconductors, b) type- l l superconductors, c )
magneti z a t i o n curve o f a type- l l superconductor
Fig.2
-
.
Structure of a s i n g l e vortex l i n e ( s o l i d l i n e : microscopic
n e t i c f i e l d h ; dotted l i n e : order parameter +).
mag-
order parameter Ji which might be i n t e r p r e t e d as a k i n d
of
macroscopic
wave f u n c t i o n f o r the condensed e l e c t r o n s , and f o r t h e vector
potential
A. They c o n t a i n two m a t e r i a l parameters w i t h t h e dimension o f
a length,
+
i .e.,
X
(Itpenetration depthl') and
5
("range o f coherence")
,
which repre-
sent the c h a r a c t e r i s t i c lengths o f v a r i a t i o n o f the microscopic f i e l d h
and o f t h e order parameter $, r e s p e c t i v e l y .
X
TypicaJ values o f
and
5
are o f the order o f some hundred Angstroms.
The r a t i o h/< i s the dirnensionless Ginsburg-Landau parameter
-II superconductors, K
I/&.
I n type-
K.
The r a t i o o f the c r i t i c a l f i e l d s Hc2/Hcl
increases monotonical 1y (but n o t 1i nearl y) wi t h
K.
The two c h a r a c t e r i s t i c
lengths and t h e Ginsburg-Landau parameter vary s t r o n g l y wi t h the mean free
path o f t h e normal e l e c t r o n s .
K
I n a c e r t a i n a l l o y system, t h e
parameter
increases l i n e a r l y w i t h the r e s i d u a l r e s i s t i v i t y l e . Therefore,
general l y are type-1 l superconductors wi t h
K
alloys
> 1/42.
I n 1957, ~ b rkosov'9
i
found p e r i o d i c sol u t i o n s o f the Ginsburg-Landau e quations which o n l y severa1 years l a t e r were recognized
e x a c t l y t h e mixed s t a t e o f type-l 1 superconductorsle.
t i o n showed t h a t t h e mixed s t a t e consist;
as
describing
Abri kosov's
o f a two-dimensional
Each f l u x l i n e can be imagined t o c o n s i s t o f a normal
w i t h a maximum o f t h e microscopic magnetic f i e l d
ji' =
array o f
4
v o r t i c e s o r f l u x l i n e s each o f which c a r r i e s one f l u x quantum
solu-
o
:
conducting
3
core
surrounded
c u r l A,
by c i r c u l a t i n g supercurrents (Fig.2).
i n an i d e a l ( r e v e r s i b l e ) type-1 1 superconductor,
these v o r t i c e s
form
r e g u l a r two-dimensional l a t t i c e , u s u a l l y o f hexagonal symnetry. The
perimental
observation o f
t h i s r e g u l a r l a t t i c e (Fig.3),
f i r s t by
a
exneu-
t r o n d i f f r a c t i o n 2 ' and I a t e r d i r e c t l y by a decoration t e ~ h n i ~ u e ~ ' ' ~ ~ , w a s
t h e p r o o f f o r t h e correcteness o f Abrikosov's s o l u t i o n s .
The
rnagnetic
s t r u c t u r e o f a s i n g l e f l i ~ xl i n e w i t h i n the l a t t i c e was a l s o s t u d i e d by a
p r e c i s e determination o f the form f a c t o r i n neutron d i f f r a ~ t i o n ~ ~ .
Because each v o r t e x contains one quantum o f magnetic f l u x ,
4
f l u x densi t y B = <h> i s r e l a t e d t o the 1i n e densi t y n o f
the
O
*
the
mean
vortices
Fig.3
-
Perfect triangular flux line lattice (FLL) as emerging from the
top face of a superconducting niobium cylinder (B = 73 mT).
The
points
of exi t of the flux lines are decorated by small ferromagnetic particles
(photograph by U. Essmann).
Fig.4
-
Irreversible magnetization curve with minor hysteresis loop.
23 1
by B =
n.4,.
I n thermodynamic e q u i l i b r i u m , t h e l i n e d e n s i t y n, and
the i n d u c t i o n B, i s constant throughout the sample, and
at
a
thus
certain
ternperature a unique f u n c t i o n o f the a p p l i e d f i e l d H ( ~ i g . l b ) : B = B ~ ( H ) .
Although the l a t t i c e parameter o f the f l u x l i n e l a t t i c e (FLL) i s severa1
orders o f magnitude g r e a t e r than t h a t o f c r y s t a l l a t t i c e s (severa1 thousand Angstroms) , there a r e many analogies between them.
The FLL i s elas-
t i c a l l y deformable, and can be described by tensors o f e l a s t i c s t r a i n and
s t r e ~ s ~ and
~ ' ~e l~a s, t i c constants connecting them27931. However, because o f the long-range i n t e r a c t i o n between the f l u x l i n e s ,
theory o f the FLL i s h i g h l y non- local.
the
This has important
elasticity
consequences
~ ~ not conf o r s t r a i n s and stresses v a r y i n g a t s h o r t d i ~ t a n c e s b~u~t 'was
sidered u n t i 1 q u i t e r e c e n t l y (Refs.24,25,34).
Another analogy
between
FLL and a c r y s t a l l a t t i c e i s t h e existence o f l a t t i c e defects (vacant li-
nes, i n t e r s t i t i a l s , d i s l o c a t i o n s , s t a c k i n g f a u l t ~ which
~ ~ ~ i ~n ~ s o
)m
cases may s t r o n g l y influente f l u x movement and t h e c r i t i c a 1 c ~ r r e n t ~ ~ - ~ ' .
2. PINNING FORCES
I n thermodynamic equi 1ibrium, the 1 i n e densi t y n o f the v o r t i c e s i s cons-
if
t a n t throughout the sample, a s t a t e which can o n l y be r e a l ized
v o r t i c e s are f r e e l y movable.
inhomogeneities w i t h s p a t i a l v a r i a t i o n s o f t h e f r e e energy
which g i v e r i s e t o a t t r a c t i v e o r r e p u l s i v e i n t e r a c t i o n
forces.
the
I n r e a l c r y s t a l s , however, there are always
The maximum e f f e c t i v e i n t e r a c t i o n force, fm,
f o r a vortex
p o t e n t i a1 s and
between
flux
one
l i n e and one d e f e c t i s c a l l e d " e l e m n t a r y p i n n i n g force". I n the f o l l o w ing, t h e main i n t e r a c t i o n mechanisrns
i n the bulk and a t
of the specimen are d e s c r i bed i n a qual i t a t i v e way.
the
More de t a i 1 s
q u a n t i t a t i v e formulations can be found i n the o r i g i n a l papers we
red to, and i n the two review a r t i c l e s of Refs. 8 and
surface
and
refer-
9.
2.1. Pinning Interactions in the Bulk
The s p e c i f i c volume o f a metal increases by about 1 0 - ~a t the t r a n s i t i o n
t o t h e superconducting s t a t e .
Thus the core o f a v o r t e x i s a contracted
c y l i n d e r w i t h i n a superconducting m a t r i x .
This e f f e c t i s f u r t h e r accen-
around
t u a t e d by the m a g n e t o s t r i c t i o n caused by the microscopic f i e l d
each vortex.
Thus each f l u x l i n e ,
e s p e c i a l l y the core b u t a l s o the r e -
g i o n o f decreasing f i e l d , i s a source o f eigen-stressesb2 which i n t e r a c t
w i t h the s t r e s s f i e l d o f a l a t t i c e d e f e c t .
This f i r s t - o r d e r i n t e r a c t i o n
( A - ~ e f f e c t , para- e l a s t i c i n t e r a c t i o n ) was f i r s t described
by
Krarner
and ~ a u e r ' and
~
f u r t h e r studied, f o r s p e c i f i c defects, by KronmÜl l e r and
~ c h m u c k e r ~5". ~
A second-order e l a s t i c i n t e r a c t i o n i s due t o the s o f t e n i ng o f the c r y s t a l
l a t t i c e i n the t r a n s i t i o n t o the superconducting s t a t e . The r e l a t i v e decrease o f the e l a s t i c constants i s o f t h e o r d e r o f 10'~. The e l a s t i c s e l f -energy o f a l a t t i c e d e f e c t depends on the e l a s t i c constants, thus i t i s
higherinandnearthecoreof afluxline.
A first
estimate
of
this
r e p u l s i v e i n t e r a c t i o n (A-E e f f e c t , d i e l a s t i c i n t e r a c t i o n ) between a screw
d i s l o c a t i o n and a s i m p l i f i e d f l u x l i n e was published by ~ e b b " ;
a
very
general treatment based on Seeger's and KronmÜl l e r ' s m i cromagnet i c equations o f superconductors47~48
was r e c e n t l y given by ~ c h n e i d e r ~ " ~ ~ .
A t h i r d , n o n - e l a s t i c , i n t e r a c t i o n i s caused by s p a t i a l v a r i a t i o n s o f su-
perconducti ng m a t e r i a l parameters li k e t h e G i nzburg-Landau
parameter
K
(hence A-K e f f e c t ) i n p r e c i p i t a t e s o r c l u s t e r s o f defects. The self-energy o f a v o r t e x 1 ine can be decreased, f o r example,
by
through a normal conducting p r e c i p i t a t e o r a void.
A
passing
partly
s i m i l a r mechanism
a c t s i n a g r a i n boundary: because o f t h e a n i s o t r o p y o f m a t e r i a l
parame-
t e r s 1i ke Hcp ( ~ e.SI)
f
and thus o f t h e Ginzburg-Landau parameter K,
self- energy o f the v o r t e x may be d i f f e r e n t i n two adjacent grains,
the
thus
l e a d i n g t o p i n n i n g e f f e c t s due t o the g r a i n boundary.
A general f e a t u r e o f a1 1 these p i n n i n g mechanisms i s t h a t they
effective if,
least o f
l o o s e l y speaking, t h e dimensions o f
the o r d e r o f the core diameter, i.e.
the defects
are o n l y
are
at
some hundred Angstroms.
Thus, defects l i k e vacancies o r non-magnetic i m p u r i t y atoms can o n l y p i n
weakl y through t h e i r eventual dens i t y f 1u c t u a t i ons, whereas a homgeneous d i s t r i b u t i o n
does n o t g i v e r i s e t o any p i n n i n g e f f e c t .
2.2. Statistic Summation
A comparison o f the observed mean volume p i n n i n g forces,
elementary
pinning
pv,
w i t h these
forces revealed t h a t the former were frequentl y one
o r d e r o f magnitude smaller than expected from a simple summation o f t h e
elementary fm forces.
This i s due t o the e l a s t i c c o u p l i n g o f the v o r t i -
ces w i t h i n the FLL which allows o n l y a small f r a c t i o n o f the p i n n i n g s i tes t o a c t w i t h the maximal f o r c e fm.
A c o n s i d e r a t i o n o f the two l i m i t i n g
cases o f a completely r i g i d and a completely s o f t FLL w i l l c l a r i f y
argument:
I f the FLL i s completely r i g i d and i f t h e r e
d i s t r i b u t i o n o f p i n n i n g s i t e s , t h e r e w i l l be an equal number
+
a c t i n g w i t h f o r c e fm,
-+
and w i t h f o r c e -fm,
this
is a statistical
of
sites
thus the n e t p i n n i ng f o r c e on
a r i g i d FLL w i l l be zero.ln a completely s o f t l a t t i c e , on the o t h e r hand,
the v o r t i c e s can occupy the p o s i t i o n s o f maximum f o r c e fm a t a11 p i n n i n g
s i tes, thus f o r a densi t y N o f t h e p i n n i n g centers, the mean w l m p i n P
The intermediate case, however, the s t a t i s t i c
n i n g f o r c e w i l l be N f
P: m'
summation o f the p i n n i n g forces i n an e l a s t i c FLL, i s a r a t h e r complicat e d problem, and i t s d e t a i l e d d e s c r i p t i o n i s f a r beyond the scope o f t h i s
article.
The f i r s t approach t o a s o l u t i o n o f t h i s problem
was made by
Yamafuji and l r i e 5 2 ' 5 3 , who considered the energy d i s s i p a t e d d u r i n g f l u x
movements (dynamic approach)
. The s t a t i s t i c a l
treatment o f the s t a t i c FLL
i n the c r i t i c a 1 s t a t e was e l a b o r a t e d by ~ a b u s c h ~ ' .For s t a t i s t i c a l l y d i s t r i b u t e d p o i n t p i n n i n g centers, both t h e o r i e s y i e l d , f o r the mean volume
p i n n i n g force, the expression
where a i s the radius o f t h e p i n n i n g centers o f d e n s i t y N B t h e magneP'
t i c induction, 0, the f l u x quantum, and S ( O ) the maximum displacement of
the f l u x a t the p i n n i n g center. The l o c a l theory o f e l a s t i c i t y o f the FLL
gives f o r t h i s d i ~ ~ l a c e m e n t ~ ~
where Ceff
i s an e f f e c t i v e e l a s t i c constant given by
Recent c a l c u l a t i o n s , based on t h e nonlocal e l a s t i c p r o p e r t i e s o f t h e f l u x
l i n e l a t t i c e , showed t h a t t h e displacement may be l a r g e r by a f a c t o r o f
up t o 100 depending on t h e f l u x densi t y B , the r e l a t i o n between S(0) and
fm,
however, remaining 1 i n e a ~ - ~ ~ .
A d i f f e r e n t approach which may be more appropriate f o r the
h i g h d e f e c t d e n s i t i e s was made by
concept o f " f l u x bundles"
56,
case
of
K o r n f e l d and K r ~ n r n Ü l l e rusing
~~
the
and f u r t h e r elaborated by Schmucker
KronrnÜl~er~"@
f o~r p i n n i n g by d e n s i t y f l u c t u a t i o n s o f p o i n t
and
defectsand
by d i s l o c a t i o n s .
2.3. Surface Pinning
I n c o n t r a s t t o these p i n n i n g mechanisms i n the i n t e r i o r o f the material,
t h e e f f e c t s o f the surface are n o t y e t so w e l l established.At verysmooth
surfaces p a r a l l e l t o the v o r t i c e s , there e x i s t s a b a r r i e r
against e n t r y
and e x i t o f v o r t i c e s r e s u l t i n g from t h e d i s t o r t i o n o f the
vortices
ne-
cessary t o obey t h e boundary c o n d i t i o n o f zero normal c u r r e n t . This d i s t o r t i o n can be i n t e r p r e t e d as a r i s i n g from "image v o r t i c e s " o u t s i d e
the
specimen, and the b a r r i e r as being due t o an a t t r a c t i v e i n t e r a c t i o n b e t ween t h e r e a l and the image v o r t i c e ~ ~ 'Roughening
~ ~ ~ .
o f the surface w i l l
decrease t h i s b a r r i e r .
Experimentally, however, i t was
frequently
ob-
served (see Section 5 ) t h a t surface p i n n i n g i ncreased when
the
~ ~ ' ~
a1~so. geometric ir r e g u l a r i t i e s
was r o ~ ~ h e n e d Thus
the surface,
l i k e asperities o r s l i p lines,
centers
of
f o r example, must be considered
surface
pinning
.
Another e f f e c t o f the surface i s the reduction o f the coupl i n g o f the
f l u x 1ines t o the FLL which can be described as a s o f t e n i n g o f the FLL
i n a surface layer. Extending Brandtls o a l c u l a t i o n s o f the e l a s t i c constants2',
t o the case o f a plane surface separating a semispace
with
a
r e a l f l u x l i n e l a t t i c e from another w i t h an image l a t t i ~ e ~i t~ i ,s possib l e t o estimate t h a t C66 i s t h e e l a s t i c constant w i t h the strongest
de-
crease i n a surface l a y e r o f 10 t o 20 l a t t i c e constants (Appendix A). As
t h e e f f e c t i v e e l a s t i c constant Ceff i n Eq.(2.2)
C66
, pinning
i s mainly
determined by
centers l i k e d i s l o c a t i o n s o r p r e c i p i t a t e s w i l l bemuch more
e f f e c t i v e i n t h i s surface l a y e r than i n t h e b u l k o f the m a t e r i a l 6 @ .
There i s no theory y e t
f o r t h e s t a t i s t i c sumrnation o f these elementary
surface and near- surface i n t e r a c t i o n s . Because o f the sof t e n i n g
FLL i n t h e surface l a y e r , however, i t i s expected t h a t the naive
of
the
suma-
t i o n o f the elementary surface forces should g i v e a r a t h e r good v a l w f o r
the observable mean surface p i n n i n g f o r c e pS.
3. THE CRITICAL STATE
irreversible
The phenomenological m d e l of the c r i t i c a 1 s t a t e describes
e f f e c t s i n superconductors u s i n g macroscopic concepts 1i k e t h e " f l u x dens i t y gradient" and the "mean volume p i n n i n g force" , w i t h o u t
considering
the d e t a i 1s o f the underl y i n g mechanisms. The r e l a t i o n between microscop i c and macroscopic d e s c r i p t i o n o f p i n n i n g e f f e c t s can be i l l u s t r a t e d by
interac-
the example o f a sand h i l l : although the elementary f r i c t i o n a l
t i o n between s i n g l e sand g r a i n s may be r a t h e r complicated, t h e knowledge
o f one macroscopic q u a n t i t y , the maximum p o s s i b l e i n c l i n a t i o n o f the s l o pe ( " c r i t i c a 1 slope"),
o f a sand h i l l .
i s s u f f i c i e n t t o p r e d i c t macroscopic p r o p e r t i e s
I n a s i m i l a r way, the cooperation o f the elementary pin-
n i n g i n t e r a c t i o n s gives r i s e t o the e x i s t e n c e o f a c e r t a i n " c r i r i c a l f l u x
densi t y gradient",
the niaximum g r a d i e n t which can be mai n t a i ned by a c e r -
t a i n d i s t r i b u t i o n o f p i n n i n g centers.
For a q u a n t i t a t i v e e l a b o r a t i o n o f
the m d e l , the e l a s t i c p r o p e r t i e s o f the f l u x l i n e l a t t i c e are e s s e n t i a l .
The l o c a l e q u i l i b r i u m c o n d i t i o n f o r an e l a s t i c continuum i n the
+
presence
o f a volume f o r c e pv can be w r i t t e n as
div
or,
-t
u + 2,
=
O
,
i n components,
where a i s t h e
(second o r d e r ) e l a s t i c s t r e s s
tensor which,
i n linear
e l a s t i c i t y theory, i s r e l a t e d t o the s t r a i n tensor E by Hooke's law. For
a t r i a n g u l a r f l u x 1i n e l a t t i c e ,
t h e tensor C o f the e l a s t i c cmstants has
t h r e e i ndependent components (e .g.,
Cll,C44
and Cs6 i n V o i g t ' s n o t a t i o d
which have been c a l c u l a t e d by severa1 authors (Refs .27-31,
34)
.
I n magnetization experiments, a two-dimensional i s o t r o p i c compression normal t o the v o r t i c e s i s the most important e l a s t i c deformation, the " bulk
m ~ d u l u s 'CL
~ b e i n g g i v e n by
The index O i s t o denote l o c a l equi 1i brium,
(aB/aH),
the r e v e r s i b l e inductions vs. f i e l d curve.
For t h i s type o f deformation,
being the slope
of
t h e components (1 1) and (22) o f the s t r e s s and s t r a i n tensors are.equal,
and a1 1 the others are zero.
The t r a c e o f the s t r a i n tensor E ( t h e
l a t i v e s i z e change o f an area S normal t o the v o r t i c e s )
the d i f f e r e n c e
fs
re-
related
to
AB o f the f l u x densi t y B from i t s equi 1i b r i u m value Bo by
Thus Hooke's Iaw y i e l d s , f o r the nonzero components o f the stress, the express i o n
I f now AB and thus the deformation v a r i e s i n a d i r e c t i o n normal
v o r t i c e s , Eq. (3.1)
-+
to
the
y i e l d s ' t h e necessary volume f o r c e pv which m s t be pro-
vided by the p i n n i n g centers t o keep the system i n e q u i l i b r i u m :
(aB/a~);'
where the symbol grad,
x-y plane norma
Friedel e t
.
AB
grad,
= B,.(BB/~H);'
. grad,
B
stands f o r the two-dimensional g r a d i e n t
t o the v o r t i c e s .
,
(3.5)
in
the
This e q u a t i o n w a s f i r s t derived
by
us i ng thermodynami c arguments , and t h e r e f o r e the e q r e s -
s i o n on the r i g h t h a n d s i d e i s sometimes c a l l e d F r i e d e l force.
~ q (3.5)
.
i s a s p e c i a l form, f o r i s o t r o p i c compression v a r y i n g normal l y t o
the v o r t i ces, o f the more general e q u a t i o n 6 6
-+
,p
,
-t
-+
= B x curl H
which may be i n t e r p r e t e d as f o l l o w s .
,
A current density
( 3 -6)
-+
= curl H
acts
237
on the system o f v o r t i c e s ( o f mean f l u x d e n s i t y
Thus, Eq. (3.6)
8)
w i t h the "Lorentz f o r -
i s the condi t i o n t h a t , i n equi 1 ibrium, the sum o f 1-orentz
force and mean volume p i n n i n g f o r c e i s zero.
These considerations g i v e the q u a n t i t a t i v e basis f o r
the
far
reaching
equivalence between the magnetic h y s t e r e s i s and the c r i t i c a l c u r r e n t
a type-l I superconductor.
of
However, whereas i n m g n e t i z a t i o n experi ments
a r e a l f l u x densi t y g r a d i e n t i s r e a l i z e d (Eq.3.5),
the deformation o f the
FLL i n c r i t i c a 1 c u r r e n t measurements i s much more complicated ( i n
gene-
6
r a l t h e f l u x l i n e s are curved ' which demands the use o f t h e more geneEq. (3.6)).
I f t h e f l u x d e n s i t y B v a r i e s continuously w i t h i n the specimen, Eq.
y i e l d s the necessary volume p i n n i n g f o r c e t o m a i n t a i n t h i s f l u x
(3.5)
density
g r a d i e n t which i s e q u i v a l e n t t o a b u l k c u r r e n t d e n s i t y . I f , on the o t h e r
hand, the 1 i m i t i n g f l u x d e n s i t y BS a t the specimen surface i s not i n equil i b r i u m wi t h the appl i e d f i e l d Ha, there i s a discontinuous jump ABS
=
-
B,(H ) o f the f l u x d e n s i t y ( e q u i v a l e n t t o a surface c u r r e n t p e r l e n a
gth o f the specimen) which can o n l s be maintained by a surface p i n n i n g
Bs
. By
s
-+
force p
-
i n t e g r a t i n g Eq.(3.5),
*
one o b t a i n s
For a given d i s t r i b u t i o n o f p i n n i n g centers, there i s a c e r t a i n
value o f the volume and surface p i n n i n g forces, pvmand psm,
maximum
which
determined by the densi t y and t y p e o f the defects, and which i n
w i l l be a l o c a l f u n c t i o n o f the f l u x d e n s i t y and temperature. I t
are
general
should
not, however, depend e x p l i c i t l y on f u r t h e r v a r i a b l e s such as the magnetic
history,
the s i g n o f t h e gradient,
the d i s t a n c e from t h e surface,
the
f l u x d e n s i t y i n neighbouring p o i n t s , e t c . I n S e c t i o n 5, we s h a l l see how
f a r these assumptions c o u l d be conf i rmed experimental l y .
A consequence o f the existence o f a maximum volume and surface p i nn i n g
forces i s the critica2 state. I t i s d e f i n e d as t h a t s t a t e i n
which
the
f l u x d e n s i t y gradient, grad,B,
and the f l u x d e n s i t y jump a t the surface,
C
values
hBS, take t h e i r maximum p o s s i b l e values ( a and a s ) . These
,,
determined by pvm and p
together wi t h Eqs. (3.5)
and
(3.8).
are
impor-
The
tance o f the c r i t i c a l s t a t e , a concept introduced by ~ e a n ' ~ i,s based on
the f a c t t h a t i n m s t experiments grad,B
themselves always t o a and
AB:.
and BSa d j u s t a n d maintain
Thus, the measurement o f these
quanti
-
t i e s gives d i r e c t l y t h e maximum volume and surface p i n n i n g forces.
Let us consider, e.g.,
an isothermal magnetization process o f a l o n g cy-
li n d r i c a l sample wi t h o u t surface pinning, b u t wi t h b u l k p i n n i n g centers,
i n a parallel field.
When the a p p l i e d f i e l d i s increased above Ecl,
l i n e s w i l l be nucleated a t the surface o f the sample.
The
flux
situation i s
unstable, w i t h consequent movement o f the l i n e s i n t o t h e i n t e r i o r , as long
as the f l u x density g r a d i e n t i s g r e a t e r than a. Upon reaching the c r i t i c a l gradient, however, the v o r t i c e s wi Il s t o p moving and
w i l l become s t a b l e .
the
cause a movement o f the v o r t i c e s u n t i l a new c r i t i c a l s t a t e i s
shed.
s i tuation
A f u r t h e r increase o f the a p p l i e d f i e l d again w i l l
establi-
Decreasing t h e magnetic f i e l d causes the f l u x d e n s i t y g r a d i e n t t o
a d j u s t i t s e l f again t o i t s c r i t i c a l value, t h i s time w i t h opposite sign.
Thus i n t h e c r i t i c a 1 s t a t e
where the upper and lower signs correspond
t o an i n c r e a s i n g and a
de-
creasing f i e l d , r e s p e c t i v e l y .
I n the absence o f surface pinning, the f l u x d e n s i t y BS a t the surface i s
i n equi l i b r i u m w i t h the appl i e d f i e l d Ha; w i t h surface pinning, B i n the
s
c r i t i c a 1 s t a t e takes the value
w i t h the same s i g n convention as f o r Eq. (3.9).
Thus the f l u x &nsity pro-
f i l e i s completely determined by these two equations. The l o c a l f l u x &ns i t y i s lower than the e q u i l i b r i u m value B,(H,)
a t a11 p o i n t s i n the i n -
t e r i o r o f the specimen i n a n increasing f i e l d , and higher i n a decreasing
f i e l d , thus leading t o a h y s t e r e t i c 5(na) curve.
As the c r i t i c a 1 s t a t e i s n o t a s t a t e o f thermdynarnic
cannot be completely s t a b l e .
equilibriurn,it
~ n d e r s 0 1 - 1 was
' ~ t h e f i r s t t o propose a t h e r -
mal l y a c t i v a t e d "f l u x creep" i n t h e c r i t i c a l s t a t e toward
thermdynami c
e q u i l i b r i u m (B=Bo). Later, t h i s e f f e c t was experimental l y observed, e.g.
i n one o f the few q u a n t i t a t i v e measurements by Antesberger and ~iimaier6B
The observed creep rates, however, a r e so low (a t y p i c a l r e l a t i v e change
o f the f l u x d e n s i t y g r a d i e n t i s o f the o r d e r o f
per hour, Refs.9 and
69) t h a t the s t a t i c c r i t i c a 1 s t a t e m d e l i s s u f f i c i e n t f o r m s t p r a c t i c a l purposes.
This f l u x creep should n o t be confused wi t h " f l u x
flow",
the movement o f v o r t i c e s caused by o v e r c r i t i c a l f l u x d e n s i t y gradients o r
overcri t i c a l currents.
The c r i t i c a l s t a t e m d e l , as described so f a r , cannot be complete i n anot h e r sense,
t o ~ :f o r geometric reasons, a f l u x densi t y g r a d i e n t normal t o
the v o r t i c e s can o n l y be created w i t h o u t macroscopic l a t t i c e
curvature
by d i s t r i b u t i n g d i s l o c a t i o n s i n the FLL3'. A simple geometric r e l a t i o n
between the d e n s i t y o f f l u x l i n e d i s l o c a t i o n s and the f l u x d e n s i t y
gra-
d i e n t was deduced and experimental l y proved by Essmann and ~ r a u b l e ~3' 6' .
The c r i t i c a l s t a t e may be i n f l u e n c e d by the presence o f these d i s l o c a t i ons, e i t h e r by the v i o l a t i o n o f l o c a l equi 1 i b r i u m i n t h e i r neighbourhood
.
o r by t h e i r i n f l u e n c e on the c r i t i c a l shear s t r e s s o f the F L L ~ ' ~ 'Recent l y , ~ c h m u c k e r 3 9 ' 4 ' t r e a t e d q u a n t i t a t í v e l y the e f f e c t o f f l u x l i n e d i s l o c a t i o n s on the s t a t i c s (Eq.3.5)
and dynamics o f a pinned FLL.
4. MEASUREMENT OF FLUX PROFILES
The o n l y d i r e c t method o f measuring elementary p i n n i n g forces fm i s p r o vided by neutron d i f f r & t i o n 7 '
s i n c e the w i d t h o f the so c a l l e d
rocking
curves i s d i r e c t l y r e l a t e d t o the mean bending o f the v o r t i c e s .
I n gene-
r a l , however, the d e t e r i n i n a t i o n o f p i n n i n g forces i n m g n e t i z e d
mens i s based on measurements o f f l u x p r o f i l e s which, by Eqs.
(3.8),
(3.5)
are r e l a t e d t o the m a n volume and surface p i nning forces
speciand
p v and
Ps.
The h y s t e r e t i c
B(H)
and rnagnet i z a t i o n curves r e f l e c t o n l y mean propert i e s
over the whole specimen cross s e c t i o n ( t h e same i s v a l i d f o r the
mecha-
n i c a l measurements,
~ef.72).
S.
This i n t e g r a t e d i n f o r m a t i o n can o n l y be
a n a l i z e d q u a n t i t a t i v e l y i n terms o f p i n n i n g forces f o r
specimens
very small dimensions i n t h e d i i e c t i o n o f the g r a d i e n t 6 8 ' 7 3 ,
with
and
without
surface pinning, so t h a t the f l u x d e n s i t y g r a d i e n t can be considered const a n t across the sample i n a good approximation.
4.1. Minor Hysteresis Loops
The method o f "minor h y s t e r e s i s loops" o f f e r s the p o s s i b i 1i t y o f measuring
separately volume and surface p i n n i n g forces as a f u n c t i o n o f
f l u x d e n s i t y i n b u l k c y l i n d r i c a l sarnples.
t h o r s f o r d i f f e r e n t measurement techniques.
the
local
I t was developed by severa1 auI n the so-cal l e d d.c.
ques, the a p p l i e d f i e l d i s v a r i e d l i n e a r l y i n time, and the
techni-
corresponding
~ ~d i ~
f f e~r e~n t i a l s u s c e p t i b i l i t y
v a r i a t i o n o f the ~ ~ n e t i z a t i oor nthe
i s measured. The same i n f o r m a t i o n i s obtained from a.C.
a small .a.c.
f i e l d i s superimposed on a l a r g e d.c.
techniques,
60'76
where
f i e l d . The v o l t a g e s i g -
na1 induced i n a pick- up c o i l can be a n a l i z e d h a r m o n i ~ a l l ~ ~d~i r *e ~
c t ~l y,
from i t s w a v e f ~ r r n ~ ~o r' ~by
~ ,i n t e g r a t i o n u s i n g a l o c k - i n ampl i f i e r s l .
A11
these methods o f minor h y s t e r e s i s ~ l o o p sare based on the generalized
con-
cept o f c r i t i c a 1 s t a t e described i n the preceding Section. As an
the technique o f d i f f e r e n t i a l s u s c e p t i b i 1i t y measurement
example,
wi 1 1 be described
i n more d e t a i 1 s i n c e t h i s technique gives a d i r e c t graph o f the f l u x
den-
s i t y p r o f i l e . The experimental arrangement c o n s i s t s o f a pick- up c o i l wound
c l o s e l y on a l o n g c y l i n d r i c a l specimen, and a compensation c o i 1 w i t l the same product (turns.area), b u t a t a c e r t a i n distance from the specimen. The
two c o i l s are adjusted t o g i v e a t o t a l v o l t a g e s i g n a l , u, which i s proport i o n a l t o the time v a r i a t i o n o f the mean magnetization
I f the a p p l i e d f i e l d H
a
v a r i e s l i n e a r l y i n time, t h i s voltage i s proportio-
na1 t o the d i f f e r e n t i a l s u s c e p t i b i l i t y
X,
Fig.4 shows schematically a h y s t e r e t i c magnetization curve o f a t y p e - l l supercondutor.
The appl i e d f i e l d i s a t f i r s t monotonously increased
t o the
a/&.
value H , and then decreased w i t h a constant sweep r a t e
The mean
o
magnetization f o l l o w s the b o l d curve connecting the two branches of thehyst e r e t i c magnetization curve.
The r e s u l t a n t induced voltage s i g n a l
o f the
two c o i l s i s shown i n Fig.5 as a f u n c t i o n o f the f i e l d decrease M.The i n i t i a 1 slope o f the minor h y s t e r e s i s loop i s (-11, equat t o t h e slope i n the
p e r f e c t diamagnetic s t a t e . This value can thus be used t o ca i b r a t e the
s u s c e p t i b i 1i t y s c a l e o f Fig.5.
The c r i t i c a 1 s t a t e model (Eqs.3.9
and
3 .10) determines u n i q u e l y
densi t y p r o f i l e B+(r) i n the specimen a f t e r i n c r e a s i n g the
the
appl i e d
flux
field
monotonously from O t o H. ( F i g . 6 , s o l i d curve). When the a p p l i e d f i e l d
now decreased, a t f i r s t t h i s p r o f i l e remains unchanged as l o n g as
B,(H,-M)
1
i s l e s s than the c r i t i c a 1 surface d i s c o n t i n u i t y
IB+(R)
is
-
AB:. The corres-
ponding h o r i z o n t a l p a r t o f the s u s c e p t i b i l i t y curve i n Fig.5
extends from
M = O t o Mo which i s r e l a t e d t o AB; by
(Appendix 8 ) . A f u r t h e r decrease o f the appl i e d f i e l d then causes v o r t i c e s
t o leave the specimen,
and again a c r i t i c a 1 s t a t e wi11 a d j u s t i t s e l f (B-(r),
d o t t e d curve o f ~ i g . 6 ) . As l o n g as the f i e l d decrease M i s n o t too large,
i.e.
B-(r)
the i n d u c t i o n d i s c o n t i . n u i t y can be considered constant,
the
i s e x a c t l y symmetric t o B + ( r ) . The rnaxirnum depth r up t o
profile
which
the
f l u x densi t y has changed i s determined by the c o n d i t i o n B+(R-x) = B-(R-x),
the i n d u c t i o n a t t h i s depth b e i ng
B+(R-X) = B, (H,)
AS
shown i n Appendix B,
-
0.5 x ( a ~ / a H ) ,
.
M, f o r AH JI AH,,
A t h e area o f the specimen's cross s e c t i o n )
and
(4.5)
(4.4)
t h e s u s c e p t i b i l i t y o f Fig.5 i s given by
when the f r o n t o f the p r o f i l e change i s a t depth x
Equations (4.4)
.
( P i s the periphery,and
.
are two l i n e a r r e l a t i o n s between
x and x
on the
Fig.5
- Differential
s u s c e p t i b i l i t y as a function o f the f i e l d
decrease
M i n the rninor hysteresis loop.
-
~ig.6
Flux densi t y p r o f i l e s ~ + ( rand
) ~ - ( r i)n the specimen during a
rninor hysteresis loop (schernatic)
.
one hand, and 'B(R-x)
arid L!H on the o t h e r hand. Thus the s u s c e p t i b i 1 i t y c u r -
ves x(M) are immediately l i n e a r representations o f the p r o f i l e B,(x)-B(H)
o o
near the surface,. turned f o r 90°, as i n d i c a t e d by the d o t t e d coordinate system o f Fig.5.
Despite t h i s important advantage f o r the a n a l y s i s o f the measured curves,
a.c.
methods a r e f r e q u e n t l y p r e f e r r e d because o f the i ncreased r e s o l u t i o n
which, e.g.
i n t h e " d i r e c t waveform analysis" o f R o l l i n s e t aZ.",
r e than an o r d e r o f magnitude h i g h e r (0.2um)
i s mo-
than i n the d i f f e r e n t i a l sus-
c e p t i b i 1i t y measurements (+- 5um). Considering t h e l a t t i c e constant o f the
FLL (0.1
-
0.2
u m i n niobium) and the f a r - r e a c h i n g mutua 1
interaction
of
i t might be doubted whether t h i s h i g h e r r e s o l u t i o n
t h e v o r t i c e s , however,
r e a l l y provides more r e l e v a n t i n f o r m a t i o n and not j u s t i n c i d e n t a l m i croscopi c d e t a i 1s.
4.2. Field Profiles on Cylinder Faces and in Slits
-+
As the normal component o f the magnetic i n d u c t i o n B i s c o n t i n u o u s a t a s u r face,
the f l u x p r o f i l e - o u t s i d e but very near a plane surface i n t e r s e c t i n g
normally the v o r t i c e s
-
i s i d e n t i c a l t o the p r o f i l e i n t h e i n t e r i o r . Three
p r i n c i p a l l y d i f f e r e n t methods have been developed f o r the o b s e r v a t i o n o'f
this field profile.
Based on e a r l i e r works w i t h a moving H a l l p r o b e 8 2 ' 8 3 , Weber
and
Riegler
measured simul taneously t h e f i e l d i n a nurnber o f very small
Hall
probes
s4
arranged a l o n g a diameter o f the sam p le . By m i n i a t u r i z a t i o n o f the p r o bes, a r e s o l u t i o n o f b e t t e r than 0.5mm was o b t a i n e d 8 5 ' 8 6 . To decrease the
e f f e c t o f t h e f i e l d d i s t o c t i o n I n the small gap between sample
face
and
probe, and wi t h i n the probe thickness, a s l i t geometry i s g e n e r a l l y used.
The o t h e r two methods can o n l y be a p p l i e d t o an end surface o f a specimen.
I n the technique o f Essmann and ~ r i i u b l e ~the
~ , p o i n t s where the surface i s
i n t e r s e c t e d by f l u x 1ines a r e decorated by very smal 1 ferromagnetic
ticles,
par-
the whole arrangement then being i n v e s t i g a t e d , by a r e p l i c a tech-
nique, i n an e l e c t r o n microscope. This method allows o f a d i r e c t
counting
o f the v o r t e x d e n s i t i e s and shows, furthermore, a l o t o f d e t a i l s l i k e d i s l o c a t i o n s i n the FLL.
However, the technique i s h i g h l y s o p h i s t i c a t e d and
time consuming, so i t s a p p l i c a t i o n j u s t t o determine
f l u x density gradi-
ents i n general does n o t seem t o be j u s t i f i e d .
A compromise between the complicated h i g h - r e s o l u t i o n and the r e l a t i v e simp l e l o w - r e s o l u t i o n method may be p o s s i b l y provided by the magneto-optical
doma i n
m e t h ~ d ~o"r i~g i~n a l l y r e s t r i c t e d t o observations o f the coarser
s t r u c t u r e o f type- l superconductors i n the i ntermediate s t a t e
gO
, but
re-
c e n t l y a l so appl ied, wi t h promisi ng r e s u l t s , t o t y p e - l l superconductorsg~.
A t h i n f i l m o f a m a t e r i a l w i t h h i g h Faraday r o t a t i o n i s evaporated
specimen surface, and the f l u x d i s t r i b u t i o n i s observed
on
a
in situ by an op-
t i c a l p o l a r i z a t i o n microscope. Because o f the use o f v i s i b l e l i g h t , i t i s
n o t p o s s i b l e t o resolve i n d i v i d u a l v o r t i c e s . However, s i n c e d i f f e r e n c e s i n
the mean f l u x d e n s i t y cause d i f f e r e n t angles o f r o t a t i o n o f the p o l a r i z a t i o n plane, f l u x densi t y gradients can be detected. This method seems
be e s p e c i a l l y a p p r o p r i a t e t o observe the general geometry
to
o f f l u x entry
and r a p i d f l u x movementsg2.
Although these methods are powerful t o o l s t o get much
f l u x d e n s i t y p r o f i les,
information
about
there remains some u n c e r t a i n t y w i t h respect t o t h e
edges o f the observed surface,
Thus they are probabl y r e s t r i c t e d
to
the
determination o f the f l u x d i s t r i b u t i o n i n the b u l k o f the specimens, below
a c e r t a i n surface sheath.
5. EXPERIMENTAL RESULTS
5.1. Flux Density Profiles
The p r o f i les, determined by one o f the methods described i n t h e
4.1 o r 4.2,
Sections
general l y e x h i b i t the f o l l o w i n g common c h a r a c t e r i s t i c s (Fig.7):
the f l u x d e n s i t y g r a d i e n t i n a. surface l a y e r o f a few pm i s extremely high.
This -i; o n l y observable by the described a.C.
methods. The o t h e r
techni-
ques determine t h i s s h o r t range drop o f the f l u x d e n s i t y as a d i s c o n t i n u -
i t y ABE a t the surface. A f t e r a t r a n s i t i o n region o f up t o 200p1, the f l u x
d e n s i t y g r a d i e n t becomes constant, o r , f o r h i g h gradients, s l o w l y v a r y i n g
wi t h the mean f 1ux densi t y .
I n terms o f the generalized c r i t i c a 1 s t a t e model as descríbed i n Section3,
the f l u x d e n s i t y g r a d i e n t ct i n the i n t e r i o r o f the specimen i s r e l a t e d t o
Fig.7
- Typicai
f l u x d e n s i t y p r o f i l e near the surface i n increasing f i e l d :
d e v i a t i o n o f the l o c a l f l u x d e n s i t y B from the e q u i l i b r i u m value B,,(H)
as
a f u n c t i o n o f the distance x from t h e specimen's surface (deformed niobiurn
s i n g l e c r y s t a l , Bo = 180 mT, T = 4.2~, Ref. 9 3 ) .
x line
o
O
Fig.8
246
-
/
/ o
o
lattice
I
O
o
Flux l i n e l a t t i c e and image l a t t i c e n e a r ' t h e surface.
o
the mean b u l k p i n n i n g f o r c e , pv, by eq.(3.5),
nuity,
w h i l e the surface d i s c o n t i -
ABz , i s caused by a surface p i n n i n g f o r c e p s (Eq .3.8). The e x i s -
tence o f a t r a n s i t i o n r e g i o n w i t h a curved f l u x p r o f i l e up t o a
distance
o f sometimes 0.2mm, however, cannot be e x p l a i n e d by t h i s model i f c y l i n d r i c a l symmetry i s assumed. Although i t was possible,
i n a s p e c i a l case
,
t o r e l a t e t h i s '8near-surface" e f f e c t t o oxides i n a surface l a y e r o f n i o g
bium ',
t h i s cannot be t h e o n l y reason because i n o t h e r experiments i t was
observed t h a t i t v a r i e d s t r o n g l y w i t h the surfaces roughness even f o r the
.
same o x i d a t i o n ~ t a t e s ~ l ' ~ ~
5.2. Volume Pinning Force
There i s a great nunber o f p u b l i c a t i o n s d e s c r i b i n g measurements
of
p i n n i n g by d i f f e r e n t defects i n the b u l k o f specimns. Somtimes,
p o s s i b l e t o r e l a t e the measured volume p i n n i n g forces
flux
i t was
quanti t a t i v e l y
the t h e o r e t i c a l l y c a l c u l a t e d microscopic i n t e r a c t i o n , mainly when
to
preci-
p i t a t e s a r e the p i n n i n g centers ( ~ e f s . 6 8 , 95-99). A q u a n t i t a t i v e i n v e s t i g a t i o n o f the i n f l u e n c e o f d i s l o c a t i o n s , however,
is
much
more d i f f i -
~ u l t ' ~ ~ This
~ ' ~ i 's . due t o the f a c t t h a t the d i s l o c a t i o n d i s t r i b u t i o n i n
deformed s i n g l e c r y s t a l s , where these measurements are made ( ~ e f s .60, 75,
102-104),
i s f a r from b e i n g homogeneous, and f u r t h e r m r e i s h i g h l y aniso-
t r o p i c , whereas the t h e o r e t i c a l c a l c u l a t i o n s are mainly r e s t r i c t e d t o the
case o f a s t a t i s t i c a l d i s t r i b u t i o n o f d i s l o c a t i o n s p a r a l l e l t o the v o r t i ces
.
Q u a l i t a t i v e l y , i t can be s a i d t h a t the p i n n i n g f o r c e o f i n d i v i d u a l d i s l o c a t í o n s i s r a t h e r weak. A q u a n t i t a t i v e d e s c r i p t i o n o f the e f f e c t s o f
an
inhomogeneous d i s l o c a t i o n d i s t r i b u t i o n , however, becomes very complicated
because o f the f a c t t h a t i n these cases besides the long-range stress f i e l d
o f d i s l o c a t i o n groups, p i l e - u p s , and networks, a v a r i a t i o n o f the e l e c t r o n
mean f r e e path i n regions o f h i g h d i s l o c a t i o n
densi t y may g i v e r i s e t o l o -
c a l v a r i a t i o n s o f the Ginzburg-Landau parameter (A-K p i nning)
. Such a com-
b i n a t i o n o f e l a s t i c and A-K i n t e r a c t i o n s i s a l s o responçible f o r the h i g h
c r i t i c a 1 c u r r e n t s o f the t e c h n i c a l superconductor NbTi. I t i s i n t e r p r e t e d
~ ~ ~ ~ ~ ~ .
as a r i s i n g from p r e c i p i t a t e s nucleated along d i s l o c a t i ~ n s ~ Possib l y , a s i m i l a r e f f e c t may be taken advantage o f t o increase
current o f A- 15 s ~ ~ e r c o n d u c t o r s ' ~ ~ .
the c r i t i c a l
For the a p p l i c a t i o n o f superconducting magnets i n f u t u r e f u s i o n reactors,
the influente o f r a d i a t i o n damage on the c r i t i c a l c u r r e n t i s e s s e n t i a l . l t
i s expected t h a t the defects produced by i r r a d i a t i o n ( i n t e r s t i t i a 1 and vacancy dislocations,loops,
voids, e t c ) a r e e f f e c t i v e p i n n i n g centers.
In
pure metals and s i m p l a a l l o y s , t h i s was r e a l l y ~ b s e r v e d ' ~ ;i n systems l i ke Nb,Sn o r Nb,Ge,
however, the r e s u l t i n g decrease o f long-range o r d e r s ,
and the corresponding decrease o f the c r i t i c a l t e m p e r a t ~ r e ' ~ ~ 'are
"~
of
much g r e a t e r i n f 1uence.
The "scal i n g laws"
first
introduced by F i e t z and ~ e b b ' " ,
and
discus-
sed e x t e n s i v e l y by ~ r a m e r " ~ , can be used t o analyse the u n d e r l y i n g i n t e r a c t i o n mechanism which gives r i s e t o an observed volume
These s c a l i n g laws are based on t h e f a c t t h a t the f i e l d
pinning f o r c e .
and
temperature
dependence o f the volume f o r c e f r e q u e n t l y can be expressed as
where the constants C and n and the f u n c t i o n f can be compared w i t h
ex-
pressions c h a r a c t e r i s t i c o f each p i nning mechanism. The exponent n, whi ch
describes the temperature dependence, g e n e r a l l y l i e s between 1 and 3, and
the f i e l d dependence can f r e q u e n t l y be approximated by an expression 1 i k e
w i t h b =B/Bc2,
O < h 5 1, and l < k
52.
5.3. Surface Forces
Although the c r i t i c a 1 surface d i s c o n t i n u i t y L B ~o r, the e q u i v a l e n t surface c u r r e n t , was observed i n a number o f f l u x p i n n i n g experiments,
n o t general l y r e a l i z e d t h a t t h i s corresponds t o a p i n n i n g f o r c e .
it
was
Further
systematic i n v e s t i gations, besides the few e x i s t i n g ones, seem t o be very
important, consi d e r i n g t h a t , on the one hand, the "surfacel'
FLL extends down t o a depth o f the o r d e r o f some
um
region o f the
(10 t o 20 l a t t i c e cons-
tants, see Appendix A), and t h a t , on the o t h e r hand, the f i l a m e n t s i n modern t e c h n i c a l composi t e wi res have diameters o f 5
-
10pm.
Most o f the e a r l i e r papers discuss the e f f e c t o f the s u r f a c e i n terms
of
an i n t r i n s i c surface c u r r e n t r e l a t e d t o the surface s u p e r c o n d u c t i v i t y b e t weenHc2andH
( ~ e f s . l 1 3 , 1 1 5 ) , o r t o image
I f these were
c3
the o n l y causes, roughening o f the surface should decrease the e f f e c t . However, the opposi t e was observed much more f requently, i .e.,
p i nni ng by rougheni ng o f the surface5'
'62.
increase
of
The surface roughness was p r o -
duced by s c r a t c h i n g o r grinding, by the s l i p l i n e s o c c u r r i n g
deformation o f s i n g l e c r y s t a l s , o r by chemical e t c h i n g .
in
plastic
I n the f i r s t t y -
pe o f measurements, i t might be d i f f i c u l t t o d i s t i n g u i s h the e f f e c t o f roughness f rom t h a t o f the d i s l o c a t i o n s produced simul taneousl y, b u t the other
two experiments seem t o be conclusive.
surface
Das Gupta and ~ r a m e r " ~observed another e f f e c t o f the surface:
p i n n i n g v a r i e d i n the same way as the b u l k p i n n i n g condi'tions were
red, even a t the same p o l i s h e d surface.
alte-
They concluded t h a t c r y s t a l
f e c t s ( d i s l o c a t i o n s i n t h e i r case) near the surface are much more
de-
effec-
t i v e p i n n i n g centers than the same species o f d e f e c t i n the b u l k . As
al-
ready mentioned i n Section 2, t h i s i s probably due t o the s o f t e n i n g o f the
e l a s t i c constant Cs6 o f the FLL near the surface (Appendix A).
A11 k i n d s o f surface p i n n i n g may be reduced considerably by p l a t i n g
the'
specimen w i t h a magnetic metal a r another supercondutor.
and
Campbell"'
~vetts"
i n v e s t i g a t e d especial l y the e f f e c t o f a d i f f u s e d t h a l l i u m l a -
y e r on the surface o f specimens o f a lead- thal l i u m a l l o y . They were able
t o suppress completely surface p i n n i n g o f unpolished as w e l l as o f
shed surfaces. The e x p l a n a t i o n i s based, on the one hand, on the
polialtered
boundary condi t ions reduci ng the e f f e c t o f the Bean-Li v i ngston b a r r i e r
57
as w e l l as o f the l a t t i c e s o f t e n i n g and, on the o t h e r hand, the d i f f u s i o n
process reduces a1so the surface
roughness
.
5.4. Field History Effects
The c r i t i c a l s t a t e model p r e d i c t s t h a t the c r i t i c a 1 f l u x densi t y g r a d i e n t
f o r a g i v e n d e f e c t d i s t r i b u t i o n i s a unique f u n c t i o n o f the l o c a l fluxdens i t y and o f temperature. The absolute value o f the c r i t i c a 1 gradientshould
thus be the same on the i n c r e a s i n g and on the decreasing branch o f an isothermal magnetization curve, and the minor h y s t e r e s i s loops should be com-
p l e t e l y symmetrical. Experiments have shown, however, t h a t both o f
requi rements are f r e q u e n t l y v i o l a t e d (Refs.76,
these
80, 1191, e s p e c i a l l y i n ap-
p l i e d f i e l d s H < 0.5 Hcp, b u t r e c e n t l y a l s o observed i n f i e l d s up t o
0.9
Hc2 ( ~ e f s . 9 3 , 120). General l y , i t i s found t h a t the c r i t i c a 1 f l u x densi t y
g r a d i e n t i s lower along the branch o f decreasing f i e l d .
No
correlation
seems t o e x i s t between t h s f i e l d h y s t o r y e f f e c t and the absolute
o f the c r i t i c a l g r a d i e n t .
c o n t r a d i c t each o t h e r .
v a l ue
The experimental r e s u l t s o f d i f f e r e n t
authors
I n the f o l l o w i n g Section, some p o s s i b l e causes
of
t h i s e f f e c t are discussed
A second r e s u l t n o t c o n s i s t e n t wi t h the c r i t i c a l s t a t e model i s an
metry o f the minor h y s t e r e s i s loops, sometimes observed i n d.c.
asym-
measure-
ments. Apparently the sharp change o f the f l u x densi t y g r a d i e n t (Fig.6) a t
r
=
R-x i s rounded o f f . Eckert and ~ a n d s t e i nconclude
~~
that f l u x varia-
t i o n s must occur even deeper than x i n s i d e the sample. I f t h i s
the case,
it
is
really
i s c e r t a i n l y due t o e f f e c t s o f nonlocal e l a s t i c i t y . For
a
complete understanding, a l s o r e l a x a t i o n e f f e c t s have t o be taken i n t o account. De
irn na'^'
observes t h i s asynunetry o n l y i f the f i e l d i s maintained
constant f o r some seconds a f t e r the f i r s t semi-cycle, before compl e t i n g
the minor loop ( t r a p e z o i d a l pulses). l f both semi -cycles f o l low each o t h e r
immediately ( t r i a n g u l a r pulses), the loops e x h i b i t a near y p e r f e c t
metry. These measurements were nade w i t h a sweep r a t e o f
sym-
he a p p l i e d f i e l d
o f 4mTIs.
When the sweep r a t e s are much higher, as they are general1
rements, r e l a x a t i o n e f f e c t s due t o the viscous damping o f the o s c i 1 l a t i n g
~~
however, t h a t t h i s e f f e c t
v o r t i c e s may occur. ~ o d m e r 'observed,
in f lu-
ences the shape o f the h y s t e r e s i s loops o n l y f o r frequencies o f about 60Hz
and higher, which corresponds t o sweep r a t e s o f the o r d e r o f a t l e a s t 100
mT/s
.
6. DISCUSSION
As o u t l i ned by ~ c h m u c k e r ' ~ ,the magnetization processes i n a
cyl indri cal
type-I l superconductor are o n l y p o s s i b l e by p l a s t i c deformation
of
the
f l u x l i n e l a t t i c e . Here, the r e v e r s i b l e p a r t o f the minor hysteresis loops
may be i n t e r p r e t e d as the e l a s t i c deforrnation o f a surface l a y e r o f
the
FLL preceding t h e . p l a s t i c flow, w i t h the f i e l d decrease Mo/2 correspond i n g t o the c r i t i c a 1 shear s t r e s s o f t h a t surface l a y e r . There
are
some
important d i f f e r e n c e s , however, between the e l a s t o - p l a s t i c deformation o f
a m a t e r i a l c r y s t a l and a f l u x l i n e c r y s t a l .
I n deformation experiments o f m a t e r i a l c r y s t a l s , the macroscopic
elastic
s t r e s s i s u n i f o r m throughout the specirnen. When the c r i t i c a l shear s t r e s s
i n the most favorable g l i d e system i s reached, p l a s t i c deformation begins
i n the e n t i r e volume.
I n t h e case o f a FLL, i n an i r r e v e r s i b l e superconductor, p i n n i n g
centers
prevent the e l a s t i c s t r e s s i n i t i a l l y from p e n e t r a t i n g f a r t h e r than i n t o a
t h i n surface l a y e r .
Thus, the surface p l a y s a dominant r o l e i n the defor-
mation o f t h e FLL. This leads t o another important d i f f e r e n c e .
The
flow
s t r e s s i n a m a t e r i a l c r y s t a l i s mainly determined by a number o f b u l k prop e r t i e s : i t i s the necessary s t r e s s t o a c t i v a t e d i s l o c a t i o n s sources o r t o
move d i s l o c a t i o n s over obstacles (e.9. p r e c i p i t a t e s o r the s t r e s s f i e l d of
other dislocations).
Although t h i s type o f f l o w s t r e s s a l s o e x i s t s i n t h e
FLL ( ~ e.39),
f
surface e f f e c t s probably p l a y here a preponderant r o l e . This
i s due t o the f a c t t h a t , i n the case o f the FLL, the e l a s t o - p l a s t i c
con-
ti nuum i t s e l f has t o be created o r destroyed a t the surface, simul taneously
w i t h the p l a s t i c deformation.
Thus, the c r i t i c a 1 shear s t r e s s ( t h e
pos-
s i b l e f i e l d decrease M o w i t h o u t h f l u x change) i s m a i n l y d e t e r m i n e d by
surface forces.
The experiments show t h a t there are a t l e a s t three sources of s u r f a c e p i n n i n g forces:
a. The Bean- Livingston b a r r i e r 5 ' a r i s i n g from image forces i s o n l y
rele-
vant a t very s w o t h surfaces o f samples wi t h weak b u l k p i nni ng, the appl i
ed f i e l d being e x a c t l y p a r a l l e l t o the surface. Otherwise,it
-
i s masked by
the much stronger c o n t r i b u t i o n s h . o r c.
b. Because o f the s o f t e n i n g o f the e l a s t i c shear constant C
66
,pinningcen-
t e r s 1i ke p r e c i p i t a t e s o r d i s l o c a t i o n s are much more e f f e c t i v e
near
the
surface than i n the b u l k (Section 2.3 and Appendix A ) .
c . There i s a d i r e c t i n t e r a c t i o n o f the v o r t i c e s w i t h the a s p e r i t i e s o f a
rough surface.
Only q u a l i t a t i v e suggestions for the o r i g i n o f t h i s
l a t t e r i n t e r a c t i o n exist
a t the mment. One c o n t r i b u t i o n should a r i s e from the l e n g t h v a r i a t i o n o f
the p a r t o f a v o r t e x w i t h i n a surface a s p e r i t y . The extreme case o f
mechanism, v o r t i c e s c r o s s i n g p e r p e n d i c u l a r l y a surface
this
we11- d e f i n e d
of
se'^'.
roughness, was i n v e s t i gated experimental 1y by Morri son and
A second c o n t r i b u t i o n t o the observed p i n n i n g o f a rough surface
may
be
due t o the n e c e s s a r i l y numerous i n i t i a l i n t e r s e c t i o n s o f a f l u x l i n e w i t h
the surface asperi t i e s . A t each i n t e r s e c t i o n ,
t o r t e d 1 2 ? which increases i t s self- energy.
the v o r t e x i s s t r o n g l y d i s -
I f the v o r t e x
it
n o t normal
t o the i n t e r s e c t e d surface, t h i s energy i s f u r t h e r increased by a
curva-
t u r e o f the 1i ne.
The depth o f the l a y e r where these mechanisms are
2vm, i n many cases o n l y about 0.5vm.
e f f e c t i v e i s l e s s than
Thus, they are no doubt
responsible
f o r the d i s c o n t i n u i t y aBs o f the f l u x d e n s i t y i n the surface, b u t they are
n o t a b l e t o e x p l a i n the observed enhanced f l u x d e n s i t y g r a d i e n t i n a d e p t h
o f up t o 200pm ( ~ i g . 7 , t r a n s i t i o n r e g i o n ) . To understand t h i s observation,
we proposed some ideasS2 which are based on the f 1 ux spot model
of
Hart
and swartz5'.
The c r i t i c a 1 s t a t e as described thus f a r i s a two-dimensional
suming s t r a i g h t f l u x l i n e s i n an i n f i n i t e p a r a l l e l c y l i n d e r .
rnodel,
as-
In reality,
however, the f l u x l i n e w i l l not p e n e t r a t e i n t o the c y l i n d e r a t once over
i t s e n t i r e length. I n i t i a l l y ,
i t w i l l be created over a s h o r t length o n l y
a t such a p o i n t o f the siirface where the n u c l e a t i o n energy i s occasional l y
lowered (e.g.
near the end faces o r a t o t h e r geometric o r
m g e n e i t i e s ) . During the f u r t h e r f l u x p e n e t r a t i o n ,
t h i s f l u x l i n e have t o move l o n g i t u d i n a l l y
( " f l u x spots")
.
chemical inho-
the p o i n t s o f e x i t o f
along t h e c y l i n d e r s u r f a c e
Even when the c e n t e r s e c t i o n o f a 1 ine i s a1 ready
b u l k o f the specimen (e.g.
i n the
200um d e e p ) , one o r both end s e c t i o n s o f
same l i n e are s t i l l i n the surface l a y e r ,
where
the
they are s u b j e c t t o the
h i g h surface forces. Because o f the 1 ine tension o f the v o r t i c e s , the deeper penetrated p a r t s o f the l i n e are coupled e l a s t i c a l l y
sections,
thus s t i 1 1 f e e l ing, although i n d i r e c t l y ,
t o these
end
the s u r f a c e f o r c e s .
This mechanism w i 1 1 be even more pronouced i f the surface i s not e x a c t l y
para1 l e l t o the a p p l i e d f i e l d .
l n samples wi t h a s t r o n g a n i s o t r o p y o f the macroscopic p i n n i n g forces (e.
g.,
p l a s t i c a l l y d e f o r m d s i n g l e c r y s t a l s ) , t h e r e i s another mchanism lea-
d i n g t o an apparent deep reaching enhanced f l u x d e n s i t y g r a d i e n t . The i n duced voltage i n the pick- up c o i l i s an i n t e g r a t e d s i g n a l over a11 azimut h a l d i r e c t i o n s . The a n a l y s i s o f the curves by Eqs.(4.4)
a n d ( 4 . 5 ) thus
y i e l d s m a n values o f the e v e n t u a l l y a n i s i t r o p i c f l u x d e n s i t y
gradients.
I f the anisotropy i s however so s t r o n g t h a t the f l u x changes
i n some d i -
r e c t i o n s have penetrated i n t o the b u l k region whi l s t , i n o t h e r d i r e c t i o n s ,
they have n o t y e t overcome the surface layer,
the
s u s c e p t i b i l i t y method
y i e l d s an averaged value between surface and b u l k p r o p e r t i e s w i t h o u t much
significance,
s i m u l a t i n g an enhanced g r a d i e n t i n a l a y e r much t h i c k e r than
i t i s i n reality.
The i n f l u e n c e o f the magnetic f i e l d h i s t o r y , the f a c t t h a t
the
absolute
value o f the f l u x d e n s i t y g r a d i e n t i s d i f f e r e n t i n i n c r e a s i n g and decreasing field,
i s h a r d l y understood y e t . One o f the p r o p e r t i e s which mightbe
d i f f e r e n t on the two branches o f the m g n e t i z a t i o n curves i s the
and d i s t r i b u t i o n o f d i s l o c a t i o n s i n the FLL (Ref.119).
If
densi t y
t h e r e a r e no
Frank-Read s o ~ r c e s 'i ~
n ~the b u l k o f the FLL, a11 d i s l o c a t i o n s have t o be
created a t the surface.
I n an i n c r e a s i n g f i e l d ,
they w i l l
o r i g i n a t e and
penetrate i n t o the b u l k together w i t h the v o r t i c e s . i n a -decreasing f i e l d ,
however, the f l u x l i n e l a t t i c e a r i s e s a t f i r s t homogeneously i n the whole
cross s e c t i o n as soon as the f i e l d reaches Hcp.
The d i s l o c a t i o n d e n s i t y
o f t h i s i n i t i a l h i g h - f i e l d l a t t i c e i s probably r a t h e r low. When the f i e l d
the necessary d i s l o c a t i o n s t o bui l d up a f l u x densi t y
i s nu4 decreased,
g r a d i e n t have t o be created a t the surface and penetrate deep i n t o the
b u l k opposite t o the d i r e c t i o n o f f l u x mvement. This w i l l i n general r e s u l t i n another d i s l o c a t i o n d e n s i t y and d i s t r i b u t i o n r a t h e r than i n an increas i n g f i e l d .
At f i r s t sigkt,
t h i s mechanism should g i v e
r i s e t o the s t r o n g e s t f i e l d
h i s t o r y e f f e c t a t f i e l d s j u s t below HcZ, where the d i f f e r e n c e i n the d i s l o c a t i o n d e n s i t y between i n c r e a s i n g and decreasing f i e l d i s g r e a t e s t ,
c o n t r a d i c t i o n t o a1 1 observations
.
At these h i gh f i e l d s , however, the shear
modulus o f the f l u x l i n e l a t t i c e i s so sma1128y24 t h a t i t behaves
l i k e a " l i q u i d l ' than l i k e a "solid",
in
more
w i t h o u t the n e c e s s i t y o f d i s l o c a t i -
ons f o r a p l a s t i c shear.
Another mchanism whích may c o n t r í b u t e t o the f i e l d h i s t o r y e f f e c t i s ba-
sed on t h e s u r f a c e b a r r i e r o f a rough surface.
s t r o n g e r f o r leaving,
I f t h i s b a r r i e r i s very much
than f o r e n t e r i n g , v o r t i c e s , i t may become p o s s i b l e
t h a t i n decreasing f i e l d some s e c t o r s o f a cross s e c t i o n a r e pinned
p l e t e l y by the surface.
com-
I n minor h y s t e r e s i s loops, these s e c t o r s w i l l n o t
c o n t r i b u t e t o t h e t o t a l f l u x change which t h e r e f o r e i s l e s s than expected
from t h e g r a d i e n t . The averaging measurement thus sirnulates a lower
flux
d e n s i t y g r a d i e n t i n decreasing f i e l d .
rninor
S i m i l a r mechanisms may c o n t r i b u t e t o the observed asymmetry o f the
h y s t e r e s i s loops. Besides t h i s , the n o n l o c a l i t y o f the e l a s t i c i t y
f l u x l i n e l a t t i c e w i l l p l a y an important r o l e here. The F r i e d e l
o f the
f o r c e on
den-
an element o f the FLL i s then determined n o t o n l y by the l o c a l f l u x
s i t y gradient, as described by Eq. (3.51, b u t a l s o by the gradients i n t h e
v i c i n i t y , up t o a d i s t a n c e o f severa1 l a t t i c e c ~ n s t a n t s ~I n~ . the model
c a l c u l a t i o n s o f Section 4 and Appendix 9, i t was assumed t h a t the
criti-
c a l s t a t e i s always r e a l i z e d by a constant f l u x d e n s i t y g r a d i e n t . By
nonlocal e l a s t i c i t y o f the f l u x l i n e l a t t i c e , t h i s cannot
t r u e i n minor h y s t e r e s i s loops near r=R-x
be
the
completely
( ~ i g . 6 1 , where the f l u x densi t y
g r a d i e n t suddenly woul d change i t s sign. Detai l e d calculat i ~ n s ' ~ ~ s h tohwa t ,
d u r i n g the f o r m a t i o n o f the f l u x p r o f i l e i n minor h y s t e r e s i s loops (Fig.6),
a generalized F r i e d e l f o r c e a c t s i n t h e d i r e c t i o n o f the sample surface
even a t a radius s l i g h t l y less than (R-x). This gives r i s e t o a b l u n t i n g
o f the sharp edge i n the f l u x d e n s i t y p r o f i le, a t r=R-x,
as observed
by
Eckert and ~ a n d tse i n 7 6 .
I n minor h y s t e r e s i s loops,
r e l a x a t i o n processes f requently cannot be ne-
glected. A t l e a s t two r e l a x a t i o n times have been observed. The f i r s t i s of
the o r d e r o f 1Oms; t h i s i s probably due t o viscous damping o f
the v o r t i -
ces, and o n l y becomes important f o r a.c. measurements with h i g h e r frequencies'".
The second r e l a x a t i o n time i s o f the o r d e r o f 1s;
it
describes
t h e f l u x creep o r " s e t t l i n g down" o f the f l u x p r o f i l e immediately a f t e r
i t was f o r c e d t o change.
This r e l a x a t i o n time i s c o n s i s t e n t w i t h the r e -
s u l t s o f d i r e c t f l u x creep measurements i n h o l l o w c y l i n d e r 6 * o r by a SQUID
technique6'.
APPENDIX A: SOFTENING OF THE FLUX LINE LATTICE NEAR A SURFACE
The p a i r i n t e r a c t i o n p o t e n t i a l between two p a r a l l e l London f l u x l i n e s
at
a distance r i s given by
U(r) = 2($,/4a
where K,(x)
)
K o(r/X)
i s the m d i f i e d Bessel f u n c t i o n , and
(A.1)
s
X
the p e n e t r a t i o n depth.
C a l l i n g J the s e l f energy o f a f l u x l i n e , the t o t a l energy o f an a r b i t r a r y arrangement o f N f l u x 1 ines can be w r i t t e n as2'
as long as a11 mutual distances are l a r g e compared w i t h core
dirnensions.
,
b u l k m d u l u s CL i s given by
F o l l o w i n g ~ r a n d t ~ ' the
w i t h the s t r a i n s
= = EYY
E
w i t h the shear angle a =
E
and the shear constant C66 by
=E,
+
E
Y"'
The two constants can be c a l c u l a t e d by
deforming the l a t t i c e homgeneously and c a l c u l a t i n g the corresponding
va-
r i a t i o n s AF o f the t o t a l energy (A.2).
E q s . ( ~ . 3 ) and (A.4)
are o n l y u s e f u l i f CL and Cs6 a r e constant w i t h i n the
volume V. To use them a l s o f o r the c a l c u l a t i o n o f p o s s i b l y v a r y i n g
cons-
t a n t s , we choose f o r the volume one l a t t i c e c e l l and s u b s t i t u t e the double
sum i n A F by a s i n g l e sum over the v a r i a t i o n s i n the p a i r p o t e n t i a l s ofone
f l u x l i n e . Takjng account o f the l a t t i c e s y m t r i e s t h i s leads f i n a l l y
the e ~ ~ r e s s i o n s ~ ~
to
Fig.9
- Softening of
plane surface.
the e l a s t i c constants CL and C,,
of t h e FLL
near
a
Considering a f l u x l i n e a t a d i s t a n c e x from a p a r a l l e l plane surface,cuta11
s i d e o f which t h e r e i s vacuum w i t h a l a t t i c e o f image l i n e s (Fig.81,
c o n t r i b u t i o n s t o the sums i n Eqs.(A.5)
and (A.6) coming
from f l u x l i n e s
deeper than 2z, w i t h i n the sample, a r e c a n c e l l e d by t h e c o n t r i b u t i o n s o f
the image l i n e s . Numerical summation becomes thus very simple and
yields
the dependence o f the e l a s t i c constants shown i n Fig.9a and b: Both const a n t s decrease considerably near the surface, e s p e c i a l l y the
shear cons-
t a n t C66, the s o f t e n i n g o f which already begins i n a depth o f 10 t o 12
X
(d.51.1m i n the case o f ~ b ) .The n a i v e a p p l i c a t i o n o f E ~ s . ( A . ~and
)
(A.61,
f o r a r e g u l a r FLL up t o the surface, would even g i v e negative shear const a n t s very near the surface.
I n r e a l i t y t h i s means t h a t i n t h i s region the
v o r t i c e s form a " f l u x l i n e l i q u i d " r a t h e r than a f l u x l i n e l a t t i c e .
APPENDIX B: DIFFERENTIAL SUSCEPTIBILITY I N MINOR HYSTERESIS
LOOPS
The f l u x d e n s i t y p r o f i l e , ~+(r),a f t e r i n c r e a s i n g the a p p l i e d f i e l d rnonotonously from O t o H
can be c a l c u l a t e d from Eqs. (3.9) and (3.10):
Decreasing now the a p p l i e d f i e l d , t h i s p r o f i
Mo, where Mo i s determined again by Eq.
sign,
(3
i 1 1 n o t change u n t i l M =
,
now w i t h
the
opposi t e
i.e.
which y i e l d s , f o r
AB:, ttie c r i t i c a 1 f l u x d e n s i t y d i s c o n t i n u i t y in t h e s u r -
face :
For
m>&!,,
f l u x l i n e s leave the sample, and the new c r i t i c a 1 s t a t e i s g i -
ven by the p r o f i l e B - ( r ) :
B- ( r ) =
B (H -AU)
o o
+ AB: + f R a . h r
~ + (,
r )f o r
r
r5R-x
,
f o r R - X ~ N R,
.
The f l u x dens i t y v a r i e s down t o t h e depth s, g i v e n by t h e c o n d i t i o n B-(RX)
= B ( R - x ) , and thus r e l a t e d t o t h e f i e l d decrease M by
+
R
so t h a t t h e f l u x d e n s i t y , a t R - x , becomes
The t o t a l decrease o f t h e magnetic f l u x i n t h e sample i s
which can be approxirnated by
i f x<<R, where P i s t h e p e r i p h e r y o f t h e c r o s s s e c t i o n A. W i t h t h e
tions
=
A ~ and
A
Ha = Ii - A i f , t h e d i f f e r e n t i a l s u s c e p t i b i 1 i t y
becomes
which cornbined wi t h ( B . $ ) ,
(8.51, and ( ~ . 8 ) ,takes the form
x
rela-
=d?/dHa
The 1 i n e a r i t y o f Eq. (B.lO)
i s a d i r e c t consequence o f the condi t i o n
x<<R.
For a c i r c u l a r cross section, however, i t i s p o s s i b l e t o g e n e r a l i z e
this
equation i n o r d e r t o make i t v a l i d f o r the whole cross s e c t i o n :
I thank my colleagues o f the low-temperature group o f UNICAMP f o r many
h e l p f u l discussions.
P i n a t t i , O.F.
able contributions.
Dr.
We are e s p e c i a l l y
grateful
t o Professor D .
G.
de Lima, A . S. B r i t o and M. S. T o r i k a c h v i l i f o r t h e i r v a l u My p a r t i c u l a r g r a t i t u d e i s due t o D r . U. Essmann and
A. Bodmer, S t u t t g a r t , f o r t h e i r c a r e f u l reading o f the manuscript and
f o r many important suggestions, as w e l l as t o D r . B. M. Kale f o r
correc-
t i n g the worst language e r r o r s .
P a r t o f the experiments described i n t h i s a r t i c l e were supported by FAPESP.
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