Magnetism and superconductivity

Magnetism and superconductivity

354 Phvsic~ 120B (1964} ~54-3(~0 Noi th -I lolhmd, Anlslentam MAGNETISM AND SUPERCONDUCTIVITY w H. MATSUMOTO, H. UMEZAWA, J.P. WHITEHEAD and G. KO...

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354

Phvsic~ 120B (1964} ~54-3(~0 Noi th -I lolhmd, Anlslentam

MAGNETISM AND SUPERCONDUCTIVITY

w

H. MATSUMOTO, H. UMEZAWA, J.P. WHITEHEAD and G. KOZLOWSKI Department o f Physics, The U n i v e r s i t y of A l b e r t a ,

Edmonton, A l b e r t a ,

Canad~ T6G 2J1

The t h e o r e t i c a l a n a l y s i s o f magnetic superconductors which i n c l u d e s , in a s e l f - c o n s i s t e n t manner, the d - f i n t e r a c t i o n in a d d i t i o n to the d i p o l e i n t e r a c t i o n is presented. The s h i e l d i n g e f f e c t of the d i p o l e i n t e r a c t i o n , the s c a t t e r i n g o f the e l e c t r o n s by the l o c a l i z e d - s p i n f l u c t u a t i o n s and the p o l a r i z a t i o n o f e l e c t r o n s by the l o c a l i z e d spins are considered. The model accounts f o r most o f the recent experimental data on a s i n g l e c r y s t a l o f ErRh,,B~.. I n t e r e s t in the i n t e r p l a y between magnetism and s u p e r c o n d u c t i v i t y extends back to the l a t e 1950's and the p i o n e e r i n g work o f Ginzburg (1) and Matthias et al (2). However at t h a t t i m e , the coexistence o f the magnetic long range o r d e r and the s u p e r c o n d u c t i v i t y could not be achieved, since the strong p a i r breaking e f f e c t (3) a r i s i n g from the s - f (or d - f ) i n t e r a c t i o n e a s i l y quenched the s u p e r c o n d u c t i v i t y . A r e v i v a l in i n t e r e s t in magnetic superconduct o r s occurred w i t h the d i s c o v e r y in 1977 w i t h the s o - c a l l e d r e e n t r a n t phenomena in ErRh4B4 (4) by Maple's group at La J o l l a and in HoMoGS8 (5) by Ishikawa and Fisher at Geneva. Although those rare e a r t h compounds (RERh4B~, REMoGS8 and REMoBSes) contain a magnetic moment, a r i s i n g from the p a r t i a l l y f i l l e d 4 f l e v e l s associated w i t h rare e a r t h ions in each u n i t c e l l , the 4d e l e c t r o n s o f Rh o r Mo nevertheless condense i n t o the superconducting s t a t e in many cases (6). One o f the most e x t e n s i v e l y s t u d i e d compounds is ErRh4B4 and a c o n s i d e r a b l e amount o f data f o r both p o l y c r y s t a l l i n e , and more r e c e n t l y , monoc r y s t a l l i n e samples (7-9) o f t h i s m a t e r i a l is available. In t h i s paper we w i l l concentrate on the p r o p e r t i e s o f t h i s compound. The presence o f the s u p e r c o n d u c t i v i t y in those compounds is due to the r e l a t i v e weakness o f the d - f i n t e r a c t i o n and i t was pointed out t h a t the dominant magnetic i n t e r a c t i o n a r i s e s from the d i p o l e i n t e r a c t i o n ( l O , l l ) , and t h a t the s h i e l d ing o f the magnetic moment by the Meissner current plays an i m p o r t a n t r o l e in the i n t e r p l a y between the magnetism and s u p e r c o n d u c t i v i t y . The notable t h e o r e t i c a l p r e d i c t i o n s based on the d i p o l e i n t e r a c t i o n are summarized below. (1) The coexistence phase w i t h the modulated spin is p r e d i c t e d and a r i s e s from the c o m p e t i t i o n between the diamagnetic nature o f the p e r s i s t e n t c u r r e n t and the ferromagnetism o f the l o c a l i z e d spins (ll). The e x i s t e n c e o f t h i s phase was confirmed by neutron s c a t t e r i n g ( 7 , 1 2 ) . ( I I ) The t r a n s verse s u s c e p t i b i l i t y behaves anomalously around i On leave o f absence from the I n s t i t u t e Academy o f Sciences, Wroclaw, Poland.

Hc~: (13). This has been confirmed by the ultrassonic a t t e n u a t i o n experiments (14). ( I l l ) The magnetic p r o p e r t i e s may be accounted for by a s c a l i n g o f Landau parameter ~' = ~/,~-T4~,~ (15} ( w i t h ;~ being the spin s u s c e p t i b i l i t y ) . This i n d i c a t e s the t r a n s i t i o n type I I / 2 .type I I / 1 type I w i t h the decreasing temperature (lO,15, 16). This behavior has been observed in magnet i z a t i o n measurement (17). As w i l l be shown l a t e r , the recent experiment (8) shows that the real s i t u a t i o n is however much more complex. (IV) The upper c r i t i c a l f l u x Bc2 is q u a l i t a t i v e i y s i m i l a r to t h a t obtained for the non-magnetic superconductors (16). (V) Since the s h i e l d i n g e f f e c t is incomplete near the s u r f a c e , the surface m a g n e t i z a t i o n can occur (18), and (Vl) the p o s s i b i l i t y o f the s e l f - i n d u c e d v o r t e x state wa~> pointed out (19). Despite the apparent success o f the d i p o l e i n t r a c t i o n model, the f a b r i c a t i o n o f a s i n g l e c r y s t a l ErRh~B,÷ in 1982 gave r i s e to a number of o b s e r v a t i o n s which did not s i t c o m f o r t a b l y w i t h i n t h i s framework. F i r s t o f a l l neutron s c a t t e r i n g studies (7) showed t h a t , in a d d i t i o n to the long wave l e n g t h modulated spin order in the c o - e x i s tence r e g i o n , there also e x i s t e d a f e r r o m a g n e t i c component, the o r i g i n o f which is s t i l l a c o n t r o versial subject. The second p o i n t was the f a c t t h a t the magnetic p r o p e r t i e s were c o n s i d e r a b l y more complex than had been supposed h i t h e r t o (8,9). I t is to t h i s p o i n t we now turn our' attention. The magnetic p r o p e r t i e s observed by Crabtree et al (8) showed a l a r g e degree o f a n i s o t r o p y . The s u s c e p t i b i l i t y measured along the a - a x i s suggested a magnetic t r a n s i t i o n temperature T~ l.OK w h i l e t h a t along the c - a x i s suggested T~ ~,-20.OK (20). The temperature-dependence of t~e upper c r i t i c a l f i e l d measured r e l a t i v e to the c - a x i s (hard a x i s ) above the coexistence temperature is q u a l i t a t i v e l y s i m i l a r to t h a t obtained in the case o f a non-magnetic superconductor. I t has a maximum value o f around lOkOe and

f o r Low Temperature Physics and S t r u c t u r a l

0 3 7 8 - 4 3 0 3 / 8 4 ) $ 0 3 . 0 0 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

Research, Polisi~

355

H. Matsumo to et al. / Magnetism and superconductivity

appears to correspond to a value of ~~14. On the other hand, the upper critical field measured relative to the a-axis (easy axis) shows not only a considerably lowered peak value but it shows Nevertheless a distinct peak at around 5.5K. Berhoozi (21) has estimated the value of K around T, taking into account the scaling effect mentioned earlier and again obtains a value of The problem then posed is, can we about 4. account for both the hard and easy axis critical field behavior in a consistent manner using the same set of parameters? Somewhat more surprising than the critical field behavior are the easy axis magnetization As the temperature is lowered, the curves. magnetization curves show an increasing convex curvature around Hc2 and as the temperature is lowered further a first order transition appears at Hc2. The fact that the first order transition at Hcl is not observed is consistent with the large K result obtained by Berhoozi. It should be pointed out that vortex pinning effects make it difficult to draw any definitive conclusions regarding the nature of the transitions at Hc~. Both of those features cannot be accounted for solely in terms of the electromagnetic interactions, which, as is shown in Fig. 1, needs

quite different K-values to fit both the easy and hard axis results. Furthermore the results presented in Fig. 1 show that the position of the peak in Hc2 in the easy axis occurs at a lower temperature than is observed experimentally. Also the transition at Hc2 is second order, since the inter-vortex interaction is modified only through x whose effects are reduced with increasing vortex density. The monocrystalline result then lead us to the conclusion that the d-f effect while in some sense small nevertheless has important consequences when we consider the finite field properties. The main effect of the d-f interaction is the modification of the electronic states by the pair breaking effect of the spin fluctuations and the Zeeman splitting arising from polarization of the localized spins. (The modification of the RKKY interaction is assumed to be small due to the effect of spin orbit coupling.) Both effects modify the order parameter A, the London penetration depth hL and the condensation energy Hg/8n. We start from the Hamiltonian

- 3-E - I$%+.i

- $ fiJo(-i$)fi

,

(1)

where the first three terms are the usual B.C.S. Hamiltonian, the fourth the dipole interaction, the fifth the d-f interaction and the last represents the residual spin-spin interaction. Due to the d-f interaction, the self energy of the superconducting electrons is modified and the gap equation in the static approximation (22) is given by A($

= 1 d3k[V - I2 t &(&E)] (*n)3 i x a ZE(g)

(2)

[tanh $ (E(~)+u) +tanh $$ (E(t)-p)],

where xs is the localized spin suscepdibility in the superconductive phase, E'(z) = c(k)'+A(rt)' and p = I. We simplify the calculation by averaging the interaction over the fermi surface and defining the effective coupling constant: g(T,n) =

-0

s

20

I

FdRk

i

3 4Ti [v - I2 ; &

(i;-t)l,;,=,+F* (3)

‘0 I IO

0

0

02

06

0.4

0.8

1.0

L

FIGURE 1 The temperature dependence of the critical fields for the dipole interaction only. Experimental points from ref. (8).

Since xs depends on temperature and the flux density n, g is a function of T and n. This approximation may overestimate the effect of the spin fluctuations at low temperatures. The gap equation (2) now simplifies to give

1 = g(T,n)N(O)

WD i

0

d&(tanhf(E+u)ttanh$(E-u)). (4)

tt. ,.1,1at,s'unzol~* c'l al.

356

Magndti.wn aim s~qmrc~mdtectirtg~

The T and n dependence o f the c o u p l i n g c o n s t a n t may be accounted f o r by means o f a s c a l e f a c t o r s ( t , n ) d e f i n e d as s(t,n)

= exp{(1/g(Tc,O)N(O)-(1/g(T,n)N(O)!.

(5)

/he s c a l e d e q u a t i o n has been s t u d i e d by Sarnla (23) and we f i n d t h a t the o r d e r p a r a m e t e r , the London p e n e t r a t i o n depth and the c o n d e n s a t i o n e n e r g y may be e x p r e s s e d in terms o f c e r t a i n u n i v e r s a l f u n c t i o n s as (22) Ii(t,n)/:';

0

:

-£(t,n)X 2

'L

'Lo

Hc(t,n)/Hco

s(t,n)~(t/s(t,n),~(s(t,n))

(6)

=~2(t/s(t,n},~/s(t,n))

(7)

= s(t,n)~b~c(t/s(t,n),~/s(t,n)),

(S)

whe re : o = 2~D e x p ( - I / g ( T o O ) N ( O ) ) -2 Lo

(9)

8 2 2 = -2e v F N(O) ,50

, o/S and

{l(i)

: t = T/T c, ~ = i#:. o. S

The f i n i t e field susceptibility ,< may be computed in terms o f net magnetic moment as ( s)-I

':HMF>/
!

~HMF / < M

7/ci

'[_ ~c(~))

(I,~,

with a, being the paramagnetic susceptibility of e l e c t r o n s and the l a s t term a r i s i n g from the shielding effect. Since the f i r s t t h r e e terms i n (13) c o r r e s p o n d t o the normal s t a t e spin interaction, we can p a r a m e t e r i z e i t as T 3 ~o + I 2,-, + 47~ = #-In _ #-D k ~ ( l + ~,~(l-cosZ,,)), where ~ d e t e r m i n e s the degree and the d i r e c t i o n o f the a n i s o t r o p y in the momentum space. When the v o r t e x d e n s i t y n i s g i v e n , the average spin polarization is o b t a i n e d from gpB J ] "M = gUBJN Bj k ~
= n.; + ((Tin/C)-4~:)
,

H(n)

4, ,,

= 7

:it,

: (n', ,,

1: "

'

and 4:rM(n)

: n-

Hin)

( Ib:,

In o r d e r to i l l u s t r a t e ti~e e f f e c t o f the f~n i t e f i e l d on the s u p e r c o n d u c t i n g q u a n t i t i e > ~ . Fig. 2 shows the dependence o f and :L a:, ,~ f u n c t i o n o f the i n t e r n a l magnetic f i e l d h( n ~ - 4 .M~). The r e s u l t s show t h a t at lower t e m p e r a t u r e s J~ is a l m o s t f l a t u n t i l h reaches some c r i t i c a l v a l u e beyond which ~ drops o f f rather abruptly. This e f f e c t is due to the Zeeman s p l i t t i n g and suggests t h a t a t lower" t e m p e r a t u r e s the e f f e c t o f the d - f i n t e r a c t i o r ; may be q u i t e d r a m a t i c . Now l e t us a p p l y the above mentioned formal i s m to ErRh~+B~, A l t h o u g h l h e p r e s e n t 4 n a l y is appears t o i n v o l v e a l a r g e number o f p a r a m e t e r . , most o f them may be d e t e r m i n e d e x p e r i m e n t a l l y The p h y s i c a l values o f the v a r i o u s p a r a l n e t e r , are l i s t e d in Table ] . The ; t r e n g t h o f the d - f

i i -d , (12)

where is the a v e r a g e mean f i e l d , ~ j is r e l a t e d t o the d e r i v a t i v e o f the B r i l l o u i n funct i o n and Jo + I2'x~ +4~ - 4 , ; } -2 L c ( l < ) / ( k ~~'+

and t h e r e f o r e ; , s ( t , n ) and g ( t , n ) may be e v a l u a t e d and the v a r i o u s s u p e r c o n d u c t i n g quailtities g i v e n i n Eqs. ( 6 ) - ( 8 ) computed. Ibis t o g e t h e r w i t h the Maxwell e q u a t i o n s and the mean f i e l d e q u a t i o n s f o r the l o c a l i z e d spins a l l o w s us to compute the f r e e e n e r g y o f the mixed s t a t e Fs(n) as a f u n c t i o n o f tile v o r t e x d e n s i t y n. The a p p l i e d f i e l d and the m a g n e t i z a t i o n may then be o b t a i n e d f o r a given v o r t e x d e n s i t y n as

Phvsi(a

~ABL~ i P a r a m e t e r ,.+ [ r R h ~ b

fc ( l cl

,,;i

[7, )

F '<2 To m

q /f

(/.>:.

T~

, -~O.uK)(?O)

i

10..,i5 (7

(I.0}

/L,)

,

, ~ .

/{

l !4

!

,4~M(=4 gi.13JN) ](). 1l !:L;e

T~

dlc(X) i dx -/T~ !,=U

0.14)* id)

(/I

=~( Tc ) N ( 0 )

{0. :,

H

( i . 4 k Oe

CO

In ( ) are r a t h e r temporar.v From the l a t t i c e structure (26)) From Tc vs x f o r LuxErl_xRh4B4 (25)

The v a l u e s

'*

interaction carl be r e l a t e d to ti~e r e d u c t i o n o f TC w i t h the c o n c e n t r a t i o n change. The n o r m a l i -

H. Matsumoto et al. / Magnetism and superconductivity

357

I

I

I

~

I

I

I

I

I

I

I 0.2

I

I 0.4

I

I 0.6

I

I 0.8

I

i

I

I

I

1.0

0,8

0,8 O

06

~

04 df

g

0.6


o.a

= 20

I2/g = 10.15

0.2

o~

0.2

=5

0 I

I

0 2

1

0.4

I

I

0.6

I

I

I 1.0

0.8

0

t

h i

i

i

i

F i

T

I

(~

-

df 5

=5

_/

~3

,,<

= 20

I2/g

= 10.15

e~

O ._J ,,<

10.15

4

=5

4

II ..E

3

f

.J ,.<

I I

I 0.2

I

I 0.4

I

I 06

I

I 08.

I

Scaling

"~" ,'<

I

B.C.S.

t = 0.6

KB = 4 df = 2 0 I2/g

l

I

..... = 0.4

I 1.0

I

02 1.0

h FIGURE 2 The e f f e c t o f the Zeeman s p l i t t i n g on the o r d e r parameter A and the London p e n e t r a t i o n depth XL. z a t i o n o f the f i e l d i s determined from Hco(Er)/Hco(Lu) = Tc(Er)/Tc(Lu) w i t h Tc(Lu) = I I . 5 K , Hco(LU)o = 1.85kOe. By means o f the r e l a t i o n H~o = ( 3 < ~ / 2 ~ ) ( ~ / X ~ o ) , the XLo is estimated as XLo ~ 825 A. The two remaining parameters, the a n i s o t r o p i c parameter ~ and d F = 4D k~/T~,are f i x e d to give a best f i t to the c r i t i c a l f i e l d s in both hard and easy axis d i r e c tions. The r e s u l t s f o r the Meissner s t a t e are presented in Figs. 3 and 4. The temperature dependence o f the condensation energy is c o n s i s t e n t w i t h the a n a l y s i s o f Crabtree et a l . The res u l t s o f c r i t i c a l f i e l d s and the m a g n e t i z a t i o n curves are presented in Fig. 5. The agreement w i t h experiments is q u i t e good. The d e v i a t i o n

I 04

I

I 0.6

I

I 08

i

I t .0

t FIGURE 3 The temperature dependence o f the o r d e r parameter and the London p e n e t r a t i o n depth in the Meissner s t a t e . o f Hc2 in the hard a x i s from experiments at lower temperature suggests t h a t the f l u c t u a t i o n e f f e c t is o v e r e s t i m a t e d , as is expected. In Fig. 6 we present the r e s u l t s obtained by neg l e c t i n g the s c a l i n g e f f e c t but w i t h s l i g h t l y m o d i f i e d parameters. There is l i t t l e qualitat i v e d i f f e r e n c e in the r e s u l t s presented in Fig. 6 and those presented in Fig. 5; in f a c t the agreement w i t h experiment is s l i g h t l y b e t t e r although the f l u c t u a t i o n e f f e c t i s now undere s t i m a t e d . Roughly speaking the dominant e f f e c t in d e t e r m i n i n g the f i n i t e f i e l d p r o p e r t i e s r e l a t i v e to the easy axis at high temperat u r e (t>O.7) is the d i p o l e i n t e r a c t i o n , however at lower temperatures (t
tl. /llatsttttt:)t:) el al. / !,lagnetis'm and ~w)urcrmductiritl

358

10

--.

B.C.S. Scaling --dr = 2 0 , ¢t = 5

(b)

....

Easy Ax s

24

~B m:C

]

4C 1 4 kOe

I ],

O8

3 555 2C

r-

O,I O

"106

-$

o

II

.I

X

4

O4

"l-

/ zr

S'

C,l o

-r

8

:y

~D O0

02

06

08

~0

t

FIGURE 4 The t e m p e r a t u r e dependence o f the condensation energy.

)2

i 10

[

r'

(Hard Axis)

O

O

^

08

t

({:) F

34

,/--li

I

i

K B = 40 c

J

l

3555

HcO = 14 kOe o 0

8

I

df = 20

[

e

I

::5

,:,

!

dE ~, c,j

4U 2C 3 555 Ol15

c. 34

I

o

r

00

04

06

4I

\

t

08

10

OJ

t

FIGURE

5

The t e m p e r a t u r e dependence o f the c r i t i c a l fields f o r the hard (a) and easy (b) a x i s , and the easy a x i s m a g n e t i z a t i o n curves f o r v a r i o u s temperatures (c). The e f f e c t s o f the d i p o l e i n t e r a c t i o n , s c a l i n g and Zeeman s p l i t t i n g are i n c l u d e d . m a g n e t i z a t i o n curve around Hc2 from a r e g i o n o f convex c u r v a t u r e (27) to a f i r s t o r d e r t r a n s i t i o n at Hc2 as the t e m p e r a t u r e is lowered ref l e c t s the i n c r e a s i n g importance Of the Zeeman splitting. The jump a t Hc2, AMII shown in Fig. 7, appears a t about t = 0 . 4 , which is r o u g h l y equal to the e x p e r i m e n t a l value (T=3.SK),

and increases l i n e a r l y w i t h decreasing temperat u r e near t r a n s i t i o n t e m p e r a t u r e T*, AMII~(T*-T). F i n a l l y i t is p o s s i b l e to determine the value o f ALo i n d e p e n d e n t l y by means o f surface impedance measurements. The i m a g i n a r ~ p a r t o f the surface impedance, ImZ :. : ~ e f f ~ B ( z ) / H ( O ) , may be e s t i m a t e d r o u g h l y as ¢ ~ ; ~ L , since i B ( z ) ~ ( I + 4 ~ x ) H ( z ) and H ( z ~ = H ( O ) e x p { - P L ( l + 4 ~ ; , ) - ' / 2 z}. A more p r e c i s e c a l c u l a t i o n r e v e a l s t h a t the non l o c a l n a t u r e o f the s u s c e p t i b i l i t y tog e t h e r w i t h the s h i e l d i n g e f f e c t r e n o r m a l i z e s the maqnetic t r a n s i t i o n t e m p e r a t u r e Tm and i n s t e a d we have Tp in the bulk and Ts a t the surface (the c r i t ) c a l t e m p e r a t u r e f o r the co-

H. Matsumoto et al. / Magnetism and superconductivity

359

0.2

(c)

(a) (Hard Axis) KB = 3.5 I

0.4

0.2

H/(C,/X

Hc0 = 1.4 kOe o

10

0.6

= 4.285

2)

o o (Easy

Axis)

0.2 K B = 3.5 \

0

T

:4.285

v

~o

-r

6

v

0.4 /

/ \ 0.6

\

\

2

0

t = 0.3

\

' 0'.4'

0

0.8 ~-

0' 6 ' 0'.8

\ t = 0.2

10

t i

2,4

(b)

0 'Z

i

i

i

i

1.0 (Easy KB

2.0

r

Axis) = 3.5

IHcO -- 4".2485Oe~

0.32

1,6

0.28

1.2

0.24

0.8

#

/

o

ix

0.4

(120 o

0v I::1

s I

0.2

I

t

I

(14

I

0.6

I

I

0.8

r-

10

i

016 o 032

t 0.08

®°° °

FIGURE 6

The temperature dependence o f the c r i t i c a l f i e l d s for the hard (a) and easy (b) a x i s , and the easy axis magnetization curves for various temperatures (c). Only the e f f e c t s o f the d i p o l e i n t e r a c t i o n and Zeeman s p l i t t i n g are included. existence phase and for the surface magnetization, respectively). In general Tm>Ts>T~ • P although Ts and Tp are ]n fact very close. The r e s u l t s o f microwave studies (28) on polycryst a l l i n e ErRh4B4 reveal t h a t ~ e f f does in fact increase as T approaches Tc2 r e f l e c t i n g the e f f e c t o f the c r i t i c a l f l u c t u a t i o n s around Ts, The value o f ~Lo obtained from these experiments

®

0.04

011

0.2

(13

0.4

0.5

t FIGURE 7

The temperature dependence o f the magnetization jump at 11c2. A refers to Fig. 5 and B refers to Fig. 6. is estimated to be around 900A which is consist e n t with the value used in the analyses o f the mixed state.

360

l[. Mat,sTring)to

('[

d/.

,~lagndlism ~ltld ~'u[~crc¢)Hd~cli]'itl

Summarizing, the p r o p e r t i e s o f s i n g l e c r y s t a l ErRh4B~, are well described by the d i p o l e i n t e r a c t i o n t o g e t h e r w i t h a weak d - f i n t e r a c t i o n . We have presented an o v e r a l l c o n s i s t e n t choice o f parameters, which e x p l a i n s both r e s u l t s o f hard and easy a x i s . The a n i s o t r o p y in the magnetic p r o p e r t i e s o f the mixed s t a t e is accounted for almost e n t i r e l y by the a n i s o t r o p y o f the s p i n spin i n t e r a c t i o n . As expected, at lower tempera t u r e , the e f f e c t o f the p a i r breaking a r i s i n g from the spin f l u c t u a t i o n s seems s m a l l e r than the present a n a l y s i s i n d i c a t e s . The e f f e c t o f the Zeeman s p l i t t i n g is i m p o r t a n t at Hc: and induces the f i r s t order t r a n s i t i o n at Hc:: observed in the easy axis m a g n e t i z a t i o n curves at low temperatures. ACKNOWLEDGEMENTS The authors would l i k e to thank Drs. M. T a c h i k i , G.W. C r a b t r e e , F. Behroozi and C.Y. Huang f o r v a l u a b l e discussions and supplying useful i n f o r m a t i o n p r i o r to t h e i r p u b l i c a t i o n s ; and Mr. R. Teshima f o r his f i n e computer work This work was supported by the Natural Sciences and Engineering Research Council o f Canada, and the Dean o f the Faculty o f Science, The Univers i t y o f A l b e r t a , Edmonton, A l b e r t a , Canada. REFERENCES (I (2

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(15) (16) (17) (18) (19)

(2o) 21) 22) 23) 24 25 (26 (27

(28

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