• * f " . • •

R p c r r " ' p IC/8T/28O

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

PHEUOMENOLOGICAL APPROACH TO THE COEXISTENCE

OF PLANAR ANTIFEREOMAG-UETISM

WITH HIGH T TYPE II SUPERCONDUCTIVITY

INTERNATIONAL

ATOMIC ENERGY

AGENCY

UNITED NATIONS

EDUCATIONAL,

SCIENTIFICAND CULTURAL

ORGANIZATION

J. Chela-Flores

A.G. Saif

and

L.N. Shehata

1987 MIRAMARETRIESTE

IC/87/280

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organizatic

INTERNATIONAL CENTRE FOE THEORETICAL PHYSICS

PIIENOMEMOLOGICAL APPROACH TO THE COEXISTENCE

OF PLANAR ANTIFERROMAGNETISM

WITH HIGH Tc TYPE II SUPERCONDUCTIVITY *

J. Ch(?la-Irlor<?R **

International Centre for Theoretical Physic:;, Trieste, Italy,

A.G. Saif *** and L.N. Ehchata ***

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

A phenomenological study In d-spatial dimensions in presented, for the

t-ooxir.icnci? of planar anttferromagnot.i cm and type II superconductivity. In

this approach critical temperatures in the; ran^e 3"-100 K arise due to the

proposed coupling of magnons and conduction electron-pairs. We discuss

some remarkable features of oxide ceramics, including a longer London

penetration depth^shorter coherence length, lower H , higher H o, large

current densities, and large pinning force.

MIRAMARE - TRIESTE

August I98T

* Submitted for publication,

** Permanent address: Physics Department, Universidad Simon Bolivar,Apartado 80659, Caracas, Venezuela and Institute Internacional deftstudios Avan^ados (IDEA), Caracas, Von^^uelfi.

*** Permanent address: Department of Mathematics and TVieoretical Physics,Atomic hrierpy K^tabliailment, "airo, \~r,::v~ •

1. INTRODUCTION

The discovery of superconductivity ( SC ) in metals and alloys, IV

in the low temperature range 0 <^ T < 30 K dates back to 1911, when

Kamerlingh-Onnes found that mercury became superconducting at 4,2 K.

It was only in 1973 when the highest critical temperature T yet reported2 C

in any metal or alloy was recorded in Nb Ge

In spite of the very low temperatures involved, practical appli-

cations are still being successfully developed in such wide ranging areas

as nuclear medicine and the Superconducting Supercollider, which will be of

great service to the high energy physics community in the 1990s.

Two theoretical approaches for constructing a theory to explain

SC in the range 0 - 30 K were possible. The first one was a phenonieno-

logical one, involving the Landau th~ ry of second order phase transitions

One of the virtues of this approach is that it describes many different

physical systems in terras of Hami1tonians of simple form, providing a basis

for universality. This was applied successfully by Ginzburg and Landau

( GL ) to the problem of low T SC ( LT SC ), providing the basis for

some important discoveries, such as the phenomenon of magnetic vortices

in type II SC . The second approach was microscopic due to the seminal

6work of Fr5hlich , followed by the successful theory of Bardeen, Cooper,

and Schrieffer ( BCS ). This theory not only was able to explain a

large body of data, but also laid down the basis for the discovery of newo

phenomena, such as the Josephson effect

The more recent discovery - in 1966 - of high T SC ( HT SC )

in the two families of oxide ceramics ( La-A-Cu-0, where A = Ba, Sr, and

Ca, as well as Y-Ba-Cu-0 ) in the well established temperature range

-2-

9, 1030 - 100 K has been Followed by reports in the higher range 100 - 300 K,

although ir is still early and firm confirmation is awaiied. Once again i.he

phenomenon of SC may be approached at two different levels: a phenomeno-

logical one in terras of macroscopic wave functions - a generalized GL method

( GGL ) , and at a more fundamental level HT SC must be understood at the

microscopic level. Several suggestions have boon put forward regarding pos-

sible microscopic mechanisms, including the proposal of Anderson , accord-

ing to which in certain ceramic structures pairs may form a rigid lattice

due Lo chemical forces. The mobile pairs playing the role of Cooper pairs

are in their lowest stale, which is ,i condensed gas of bosons.

Many different proposals have been made since Anderson's work,

ilowever, it i:: si ill too early to choose amongst these mic rosr.opic theories.

The reason is that key experiments are still to be confirmed. A typical

illustration concerns the isotopic effect: It is generally agreed that:

18 16subst i [ ut ion ol 0 for 0 causes no change in T , but oxygen subst ilu -

I ion in the La-Sr superconductor has been recent ly reported t.o lower T by

1 2 '"about I K . W e must wait therefore for clarification of this crucial

experiment on which microscopic theories must be based.

This slate of affairs has led us to lake a new look at the rele-

vance of simple phenomcnologica1 proposals. A few suggest ions have been

advanced as It) how to proceed in establishing ,i useful GI. formulation

GInzburg suggested that in the free energy expansion two modifications

should be introduced - mass anisolropy and inclusion of | *1) | terms

1 4Baskaran and Anderson , on the other hand, suggest lo consider a U( 1 )

gauge theory of HT SC on the lattice. With doping the coherence length

increases, and when ii is greater than a few lattice paramelers it is

appropriate to rr.pl,-ice the free energy by its coarse-grained version,

which is then shown to be equivalent to the GL free energy, A third phe *

nomenological suggestion proposes to understand a GL FormaLism in terms

of two order parameters corresponding, respectively to two different types

of electron pairs in the supereondueting and insulat ing states -

In spite of the above preliminary investigations, we feel that

two deep aspects of the problem have not been fully understood:

( i ) The role played by planar ant iferromagnet ism ( AF ) in enhancing

T beyond the range 0 - 30 K, and

( ii ) The proper inclusion of the renormalization group ideas, so as to

have a theoretical basis wider than Che GL formal ism

With respect, to point ( ii ) above we should further recall that.

in general critical phenomena are best studied in d-spatial dimensions.

V -4

In the specific case of LT SC the coherence length C - N / 1 0 cm, and thus

the Landau theory is we LI suited t.o make reliable p re diet ions in the

physical space of dimensionality d - 3, On the other hand, in HT SC the

new coherence length y is much sma Ller than the old one - about 20 A

as experiments have shown. In this new situation the GL formalism cannot

be expected to hold without modifications.

2. TOWARDS A d-DlMENSIONAL MODEL FOR HIGH T SUPERCONDUCTIVITY

( a ) The need to go beyond the Landau theory of critical phenomena

The convenient theory of Landau provides only a first stage

in the description of critical phenomena, since analyticity of the rhermo-

dynamic variables is assumed, and the exact solution of the Ising model

of ferromagnetism ( FM ) showed that the partition function is nonanalytic

1 Rat T . This il lust rates in parr irular thai r he analyL icity assumpt ion

of the GL |-henry must be handled care fully. The re normal izatLon group

r* 19and Q , -expansion ( i.e. , d - 4 expansions ) provide a framework for

more reliable description of HT SC.

In Sec. 1 we discuss a CGL mode I in which due care is paid ic d

and. n , who re n denote s the dimensionality of i he order parameter. How -

cvur, before1 we do so we should become more acquainted with the nalure

of : 'ie order paramo. I er in self: La,CuOr is the parent compound of the

2 a - yd o p e d HT SC o f t h e t y p e La A CuO , a n d e x p e r i m e n t s s u g g e s t t h a i b e l o w

c 2 - x x 4 - y

t h e Nee 1 [ empe r a f u r t i T o f a b o u t 2 3 0 K t he r e i s a c l e a r i inoma ly i n t h e

roagne r i c s u s c e p t i b i l i t y , f o r v a 1 uc s o f x r a n g i n g [ roni 0 . 0 0 5 t o 0 . 0 0 6 5 ,

w h i c h i s a 1 r e a d y l o s t . f o r x - 0 , 0 1 . T h i s imp l i e s t h a t 1 h e l o n g - r a n g e AF

o r d e r i s l o s t , by o n l y 170 o f i m p u r i t y on t h e l a n t h a n u m s i t e s . The so. e x p e r ' -

!(ir--: \ I:-, i n d i c a t e t h a i rip i n d e n s i I y w , w c s ( SDW ) must" p 1 .a y a 1 e a d i ng r o l e

in HT SC. T h i s w i l l be t h e t o p i c : o f ( h e n o x l s u b s e c l i o n .

{ b ) T h e p h y s i c a l p i c t u r e u n d e r l y i n g t h e p h e n o m e n o l o g i c a l a p p r o a c h t o

p l a n a r a n t i f e r r o m a g n c M . i s m .

We may a s s u m e t h a t b e 1 ow T SDWs may p r o p a g a I e wi I h a w . i v e _ v e c t o rN

k, su t ha t a sma11 exc it at ion one rgy E may be L aken as t he sum of (he

4- S

i nd i v i dua 1 magnon enorgie s r L^5' » who re CxJ • dcnoics the frequency of the

SDW of ! ype j :

(2.1)

Hc'c n j ' i ' denotfs Che number of SDWs with w,ivu-vector k.

For T S X T N »e have neglected Interactions ( i.e., Eqn. (2.1)

represents a magnon gas ). We may assume Chat SDWs condense into the

state j with lowest energy:o

(2.2)

since it is possible in AFs that several SDWs with the same wave-vector Q

may propagate . The use of macroscopic wave functions is then possible.

( c ) Derivation of the order parameter for planar ant iferromagnetism

The ground state of the AF system will have a magnetization in the

X-Y plane, which changes sign from one unit cell to the next. Dividing

the lattice into two sublattices A and B we may define the variable

K- =where ^. denotes the spin angular momentum on the ith lattice site, and

sub lattice AI = +1, on

£_ • - -1 , on sublattice B

(2.1*)

(2.5)

( In ref. 23, the variable K. is denoted by Q.. We have avoided this

notation in order to reserve thefvector Q for the wave-vector of the

SDW-condcnsate ) . In the present work we prefer the macroscopic magnon

theory, and choose the "staggered" ( i.e., spread over the two sublattices )

magnetization density field N( £ ), rather than the individual spins K.

-6-

d e f i n e d o n c i t h e r l a t t i c e :

Ktt ( x _ X ^ , (2.6)

where r he U -funor. ion spreads out in space inc. 1 uding a silo from o.ich sub-

lattice, and where X J denote s the it h sp in. Tn terms of the vector N we

may define t he sub 1 at r ice magnet izal ion in the X-Y plant1 ( i.e., t. he order

paramct or ) :

(2.7)

w h o r e w o h a v r u ^ n d t h e d - d i m e n s i o n a l n o L a L i o n x ( r , r , . . . , r ) . T h e

m a g n i t u d e o f t h e s t a g g e r e d t n a g n e t i z a t i o n [ Q j | i s o f m n w r i t t e n a s

N ( x ) . I n I h e d e f i n i t i o n o f t h e o r d e r p a r a m e t e r ( y C X ) a s g i v e n b y

E q n . ( 2 . 7 ) , t h e d i m e n s i o n n o f t h e o r d e r p a r a m e t e r i s e v i d e n t l y n 2.

( d ) T h e CGL mode l , f o r p l a n a r a n t i f e r r o m a g n c t i s i ^ .

We c o n s i d e r nex t , a GGL m o d e l f o r p l a n a r AF i n d - d i t n r s i s i o n s , i n

7kw h i c h ( h e e f f e c t i v e H a m i l L o n i a n i s f ; i v e n b y :

dL

(2.8)

w h e r e t h e q u a n t i t y ( ^ d c n o l O S , a s i l d i d i n S e c . 2 ( b ) , l h e w . i v c - v e r . t o r

-7-

of the Lowest energy SDW Into which the spins condense for T smaller than

T ( c.f., Eqn. (2.2) ). We have retained d as a free parameter in orderN i /i Q /

^ and 1^to be able to discuss the region where Q^ and 1^ are expected to be

nonanalycic ( cf.t Sec. 1 ); we have limited our discussion to the case

n = 2 in order to enforce the criterion ( i ) in Sec. 3{b).

3. GENERALIZED G1NZBURG-LANDAU EQUATIONS

( 3 ) The general problem of the coexistence of antiferromaftnetism and

superconductivity.

The coexistence of SC and magnetic ordering has been studied for

some time now , particularly regarding the Chevrel compounds of the class

MMo,Xn ( M = rare earth, X - S or So ), and MRh E,. In these compounds SC

may I urn into FM. In fact, in ErRh B as the temperature decreases there4 4

occurs a second order phase transition to SC with T - 9 K., then therecl

is a first order transition to FM at T = 0.9'i K. This "reentrant" SC

has been studied in a GGL theory in terms of two order parameters ( the

macroscopic wave function and the magnetization ), and thus there arc some

ideas in common with our work. However, the reentrant GGL theory refers

to the coexistence of FM and LT SC in the Chcvrel compounds, whereas oursc r

refers to the coexistence of planar AF and type II HT SC in oxide ceramics.The first experimental confirmation of the coexistence of SC and AF occurred

27in 1977 in the compound MMo S ( M = Gd, Tb, and Dy ) , and later in

28 6 8

the rhodium-boride phase SmRh,B . The approach of some previous theories,

are based on the remark that, when AF and SC coexist, low-lying collective

29excitations associated with these coexisting long-range orders interact

with a coupling constant (i at the appropriate temperature domain < LT SC

for the Chevrel compounds and HT SC for rhe oxide ceramics ). Such interaction

c

leads in g neral - as we will confirm below -to appreciable changes in rhe

physical quantities, such as A i t , I , just lo name a Few ( cf., Sec. 3d ).

L r JC

( b ) The effective HamiIranian.

In order to construct an ansatz for [he effective Hamiltonian Heff

of t h e GGL a p p r o a c h , we a r e g u i d e d by [ h e f o l l o w i n g c r i t e r i a :

( i ) T h e o r d e r p a r a m e t e r s s h o u l d h a v e I h e same d i m e n s i o n n , a t l e a s t i n

I he l i m i t i n w h i c h a n i s o t r o p y i s n e g l e c t e d ,

( i i ) T h e s p a c e d i m e n s i o n a l i t y i s d ,

( i i i ) T h e e f f e t l i ve H a m i l t o n i a n must be a . s c a l a r , a n d

( i v ) T h e t h e o r y mus t be £ a u g o i n v a r i a n t .

We h a v e s e e n i n S e c . 2( c ) t h a i f a r p l a n a r AF t h e d i m e n s i o n a l i t y

of [ h e o r d e r p a r a m e t e r n - 2 . I n f a c t i t i s k n o w n t h a t [ h e I u o - c o t n p o i i c n t

s t a g g e r e d m a g n e t i z a t i o n o r d e r p a r a m e t e r i n E q n . ( 2 . 7 ) i s t h e o n l y f o r m

<>A AF w i t h n - 2 . T h e n - 2 o r d e r p a r a m e t e r of SC may b e w r i t t e n a s :

f i lm: a i l i s k n o w n i n HT SC t h a i I h e f 1 ux qurint i z a t i o n

e l e c t r u n s a r c p a i r e d

(3.1)

I hal i he

( c ) The e q u a l ' i o n s o f t h e GGL f o r m a 1 i s m .

Tn o x i d e c e r a m i c s SC a n d AF may c o o x i.si a n d be u n d f r s [ o o d i n

terms of an ansatz which cannot be derived from first, principles, but is

obtained from the effee Live Hamiltonian:

#;>.

where

(3.2)

(3.3)

(3.4)

5.5)

Tn these formulae, \ D and m are the order parameters associated

with SC and AF respect, ively; h = curl A is the internal magnetic field, and

A is the vector potential; Q was defined in Eqn, (2.8), and 0 is the

coupling constant. In general the constants dk p ()> Q VA ij" and "jj

are temperature-dependent.. The Hamiltonian ( 3,2 ) - ( 3.5 ) may be

minimized with respect to the variations in (T) (TJ K a n d Q. This

leads to the four GGL equations:

-10-

-The new coherence length. Range of variation of the order para-

meter w : In the case of a very weak field inside the superconductor,

\XJ is expected to vary very slowly, close to the value \ \^M , where

^ T j _ \ a j \ £ . The range of variation of \ij can be deduced from the first

GGL equal ion ( i.e., (3.6) ) , by letting A - 0. In this case the con -

I ribut ion of the wave-vector Q can be neglected ( cf,, Eqn. (4.3) below ).

Using Eqn. (3,6) we obtain:

The new coherence length is related to the LT SC coherence

length r , where £ ^ _ ^ fjlXfl Cil by 'ho rcla! ion,

(3.13)

From Eqn. (3.14) it may be easily shown [hat the new coherence

length V -. f when the coupling constant *Q vanishes, and/ or

when \A •=. 2. Oi. rrii . In addition, when ~\ - " & c , lj •= O • This

mrjns rh.it: in [he interval — " o ^ \ u <^ Q L he coherence length V de-

creases. As Q becomes positive JT monnf onica 1 1 y increases up

to infinity at <• c f - F i S - 1 ) •

Ci.li)

-Critical current in a thin film or w i r e : The G G L e q u a t i o n s

can be applied for c a l c u l a t i n g the cricic.il current in a thin super -

conduct ing sample at w h i c h the SC may break d o w n . When the sample d i m e n s i o n

is much smaller than \ and jt , \\J^I and J are expec t e d to be a p p r o x i -

m a t e l y constant over the sample c r o s s s e c t i o n . By setting \1J a c c o r d i n g

-1.3-

to Eqn. (3.1) (and keeping | V J | independent of x ) , we find that when

there, is no magnetic field Eqns. (3.7) and (3.2) yield:

u; =, (~k?s_ ^ A") , (3.i5)where we have used the relations VLl=.\u;\3 and \Vl/\_ ei I 6 •

For vanishing coupling constant J reduces to the critical current of LT SC

Moreover, when 15 i —"l^.-s. — (Q.fy\*|^S" and/ or -$ = ^ the critical current

"^ ,»O • However, J reaches its maximum value when -5- = 2/3,

i.e., the critical current density is:

(3.16)

There is ample experimental evidence in favour of Eqn. (3,16). Some

preliminary expe riment s point towards important Lechnological irnplicaLions

in ceramic superconductors: Wires and thin films have been fabricated and

2 2a large critical current density of 7,25 x 10 A/ cm at 77 K has been re -

pori.cd . There are also reports of current capacity exceeding 10 A/ cm

at. t hp temperature of liquid nilrogen; this is sufficient, for powe r trans -

mission. In face, critical supercurrent densities are reported as 3 x 10

A/ cm in low fields at 4.5 K, remaining above 10 A/ cm for fieLds as

large as 40 kG

-Flux quantization: This is a general feature of SC. It may be

inferred by using Eqn. (3.7) and assuming Eqn. (3,1). Proceeding as usual32

we find:

-A -

( 3 . 1 7 )

T h e l i n e i n t e g r a l o f A a r o u n d a c l o s e d l o o p \ g i v e s t h e f l u x o i F\_ t h r o u g h

t h i s l o o p :

r(3.18)

W h e n a f u l l [ u r n i s c : o m p l e t e d a r o u n d t h e l o o p \ , i n d s i n c e r h e t w o f u n c t i o n s

V^J a n d m m u s t b e s i n g l e - v a 1 u c d , I h e i r p h a s e s v a r y h y l T ^ f l a n d 1 1 3 1 K j

r e s p e c t i v e l y , w h e r e n a n d k a r e two n o n v a n i s h i n g i n t e g e r s :

< i. ] 9 >

By m e a n s of E q n s . ( 3 . 1 8 ) a n d ( 3 . 1 ° ) w e t: ail w r i t e E q n . (") . ! 7 ) , .if t f r A line

i n t e g r a t i o n , a s f o l l o w s :

w h o re

r ^ 2 x 1 0 g , i u s s . cm '

(3.20)

(3,21)

and,

£ = A / n (3.22)

In Eqn. (3.21) ^ denotes the flux quantum in 1,T SC, and in Eqn,

(3.22) £ is some parameter dependent on Q and | UJ | ( in these deriva-

tions the path of integration 1 for the current was taken to be a line

where J = 0 ). Now, > = 0 when 0 = 0 and there is no change in the flux

Vet, S> may change as the parameter £ changes according to "3" •

h. DISCUSSION AND CONCLUSIONS

In the region of validity of the GGL model developed in Sec. 3

SC and AF coexist. Therefore magnetic order does not affect, the SC Kubstan-

36tiull.v, and we expect

where k denotes the Fermi wave-vector. This may be understood by

considering the extreme case

Q < < ( 4 . 2 )

Tn this regime SC must be influenced by AF since the presence of magnetic

ordering effectively scatters the Cooper pairs of those conduct ion electrons

close to tho Fermi surface. These limits have been used in Sec. 3. [n

passing wo would also like to point out thai: in the case of Eqn, (4,2),

the dominant wave-vector of the spin fluctuations that- contribute to the

scattering of the conduction electrons is

0 (4.3)

-16-

The main assumption of Sec. 2( a ), namely the ex isl c-nce of

SDWs, has received experimental confirmation at least in the related

phenomenon of LT SC in the Chevre! compounds. In facl, neutron experi -

ments have confirmed that the associated magnetic structures are ( sinu-37

soidal ) SDWs . In the quasi-one-dimensional, highly anisotropic organic

structures ( TMTSF ) X, measurements have also shown the coexistence of SDWs

and SC.

On the other hand,the electrodynamics of HT SC nre understood in

the framework developed in Sec. 3, and it. must be underlined that from

magnetization measurements there are strong hints that the cI eelrodynamica1

39properties of HT SC is that, of conventional type II superconductors . The

GL p a r a m e t e r c i 2 = . A / 5 f o r t h e s e t y p e s <if m a t e r i a l s i s a l w a y s m u c h

g r e a t e r I h a n u n i t y . T h i s i s r e a l i z e d i n o u r c a l c u l a t i o n o f t h e n e w h i g h e r

X a n d t h e n e w l o w e r £ . T h e f i r s t c r i t i c a l f i e l d R c ~ o 3 ? - C r \ J i s n o w

d e c r e a s e d ( i . e . , w e h a v e a r e d u c e d M e i s s n o r s l . i t e ) . T h e s e c o n d c r i t i c a l

H t ~ 40

c , l " " / . J i s i n c r e a s e d , a s e x p e c t e d from measurements

g e n e r a l t h e thermodynamic c r i t i c a l magne t i c f i e l d H t - s j

f o r t h e new type I I c e r a m i c s u p e r c o n d u c t o r s i s i n c r e a s e d . Our c a l c u l a t i o n s

a l s o s h o w c o m p l e t e a g r e e m e n t , w i t h t h e m e a s u r e m e n t s o f I h e f l u x q u a n t u m , a s

w e l l a s w i t h t h e c r i t i c a l c u r r e n t c a r r i e d b y t h i n f i l m s o r w i r e s . I t i s

e a s i l y s e e n f r o m E q n . ( 3 . 1 6 ) t h a t t h e n e w c r i t i c a l c u r r e n t f o r H T S C

c o i n c i d e s w i I h L T S C w h e n J = O . H o w e v e r , w h e n Q = — ~& ~\ i-vc t ~J^ — *-•

as expected. Since the new coherence length decreases, I he flux pinning

force F -v^ 2C i s expected to be lowered

We should next clarify the range of values of the physical parameters

for which our phenomeno 1 ogica 1 equations .sre expected to produce reasonable

In

^ \]^

-17-

agrecment with experiments. These parameters are:

( i ) T - the mean field theory value for the experimental criticalo

temperature T ,

C ii ) y - the oxygen defficiency parameter, and

{ iii } x - the doping parameter.

Regarding the first case, just as It was in the GL theory, the

range of validity with respect to temperature is:

T - T <V" T (4.4)

o ^ ^ o

However, we should recall that the condition (4.4) was originally imposed

so as to guarantee that the order parameter was small, and that,

(4.5)

for a pure metal. Yet., it has been pointed out that far Sn ( T = 3.72 ),

43I T - T I - 0.1 K . Therefore, we feel that we should consider carefully

o

t.he region of applicability of the theory, particularly with regards to x

and y, as discussed below.

Regarding the second case, the parent compound La CuO is anti-

ferromagnetic provided y is greater than a few per cent, thus -T = 0

for y = 0, but T is aLready 295 K for y = 0.03 . This result is clear

from magnetic susceptibility anomalies present for 5maLl y, but disappear-

ing beyond some larger value of y ( close to 0.5 ). Therefore, the existence

of a small oxygen vacancy concentration appears to be necessary for in -

duc.ing AF in the parent compound La CuO ( cf. , Fig. 2 ).i. 4—y

-18-

With respect to case ( iii ), we know thai a magnetic suscepti-

bi1ity anoma Lv appears for t_he compounds ( L a Ba ) CuO for x = 0,1-x x 2 4-y

0,005, 0,OO65» but already disappears for x ^ 0. 1. In Fig, 3 we have illus-

trated with simple linear interpolation the behaviour of T and T ,

From what has been said above we have understood better the range

of validity of our phenomeno Logical approach, havi ng suggested possible

ranges of values of x and y for which HT SC and AF may coexist. However,

it should be realized that the illustrations are possible at this stage

for the Ian t hanum compounds * It: is conceivable that the description of

the y11 r ium compounds may requ i re some adjustments, even some modi fieations

of the phtnomenological descript ion.

Keeping these limitations Ln mind, in restricted ranges the values of

the oxygen dcfficicncy paramcter y the theory may be expooLed to yield

reasonable predict, i ons : This is illustrated in Fig. 2 for the lanthanum

compounds ( t: f . , shaded region ). Inside the rectangle ,

.6)

where y is the value of y for which,

T = T = Tc N o

a n d E q n . ( 4 . A ) may 1 h e n bo g u a r a n t e e d t o h o l d f o r t h e m e a n - f i e I d - : h e o r y

critical temperature T .

o

Similar rest ricted ranges for i.he doping parameter x may be

obtained: This is illustrated In Fig. 3 ( cf., shaded region ) . Inside

(4.7)

-19-

the rectangle . & X we have

\ X o \ (4.8)

where x is the value ofthe doping parameter for which Eqn. (4.7) holds,

so that the validity of Eqn. (4.4) may be guaranteed for the single

mean-field-temperature T .r o

To conclude we would like to make two final comments. The first

one concerns the heuristic value of our simple linear interpolations ( Figs.

44,452-3 ). The trends learnt from careful microscopic calculations do

support the main point, we wanted to illustrate schematically with the

diagrams (at least for the lanthanum compounds ), namely that in restricted

domains of T , x , and y the above GGL formalism, based in the coexistenceo o o

of AF and SC presents us with a valuable framework for discussing the

experiments available, and for making new predictions.

The second comment concerns Lhe value of doing phenomenology,

rather than microscopic theories. It is well known to any experimentalist

that has worked in LT SC that, the GL formalism is of enormous heuristicc

value . As pointed out in Sec 1, the whole field of type II SC arose

from careful consideration of the magnetic field responses of metals

and alloys, within the framework of the GL model. For these reasons we

believe that at. this stage the GGL formalism developed above may provide

some insights into HT SC.

-20-

ACKNOWLEDGMENTS

The authors would like to thank Professor Abdus Salam, the

1 n! ernat ionaI Energy Agency and UNESCO for hospitality at tlie Interna-

tional Con! re for Theoretical Physi.cs, Trieste, The work was done while

one of [he authors (J.C.F.) was holding a grant from Fl Dlpartimento per

la Coope r.iz i one .i 1 1 o Svilippo del Minislero dtJgli Affari Fstori ( Italy ) .

We would also like lo 1 hank Professor N. H. Narch lor his encouragement

and a cril iral reading of the manuscript .

-21-

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FIGURE CAPTIONS

Figure 1: Schematic rcpresentat,ioti of the dependence of the square

of the new penetration depth A. ( I ), and the square

of the new coherence length t ( II ) on the coupling

Figure 2: The range of values of the y parameter ( oxygen defficiency )

for which the critical temporal lire T and I he Neel temperature

T lie in a small domainN

where T ~ T - T . The value<: N o

of y 0 .006

f i g u r e 3 : The range of v a l u e s of the x - p a r a m e u - r ( dop ing p a r a m e t e r )

for which T and T l i e in a smal l f e m p e r a [ u r n domain

where T - T T . The value of x - 0.0088. SC and AFc N o o

coexist inside the triangle OAB.

-26-

•m^-M: 1:

Fig.2

Fig. l

295K

10

Fig-3