INDUCED MAGNETISM IN AMORPHOUS RARE-EARTH ALLOYS

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INDUCED MAGNETISM IN AMORPHOUS RARE-EARTH ALLOYS

A. Bhattacharjee, B. Coqblin

To cite this version:

A. Bhattacharjee, B. Coqblin. INDUCED MAGNETISM IN AMORPHOUS RARE-EARTH AL- LOYS. Journal de Physique Colloques, 1980, 41 (C8), pp.C8-626-C8-629. �10.1051/jphyscol:19808157�.

�jpa-00220257�

JOURNAL DE PHYSIQUE CoZZoque C 8 , suppZ6rnent au n08, Tome 41, aodt 1980, page C8-626

INDUCED MAGNETISM IN AMORPHOUS RARE-EARTH ALLOYS

A.K. B h a t t a c h a r j e e and B. Coqblin

Laboratoire de Physique des soZides+, Uniuersite' Paris-Sud, Centre dlOrsay, 91405 Orsay, France.

ABSTRACT

A s i n g l e t - s i n g l e t model f o r induced magnetism i n amorphous systems i s developed by assuming a g a u s s i a n d i s t r i b u t i o n of t h e e n e r g y s p l i t t i n g . The s u s c e p t i b i l i t y , t h e spontaneous m a g n e t i z a t i o n and t h e s p e c i f i c h e a t a r e computed i n a m o l e c u l a r f i e l d a p p r o x i m a t i o n and a r a t h e r s a t i s f a c t o r y agreement i s found w i t h e x p e r i m e n t a l d a t a i n P r amorphous a l l o y s .

I - IdTR3DUCTION

Recent experiments 11-51 on amorphous a l l o y s c o n t a i n i n g Praseodymium have shown t h e importance of t h e s i n g l e t c h a r a c t e r of t h e ground s t a t e of t h e Pr i o n a t low t e m p e r a t u r e s . Magnetic measurements i n amorphous Pr Ag a l l o y s ( w i t h x = 50 and

x 100-x

21.5) s u g g e s t t h a t a b o u t 30% o f Pr i o n s a r e i n a s i n g l e t ground s t a t e a t 4.2 K [ I ] . The magnetic s u w c e p t i b i l i t y x of d i l u t e PrxLa80-xA~20 a l l o y s follows a Curie-Weiss law above 100 K , w h i l e t h e low tempe- r a t u r e s l o p e of 1 / x v e r s u s T c u r v e i s t w i c e a s much a s t h e h i g h t e m p e r a t u r e v a l u e and t h e s a t u r a t i o n magnetic moment i s r o u g h l y 1.4 pB [ 2 ] . An a n a l y s i s 131 of t h e s e d a t a s u g g e s t s t h a t r o u g h l y 50% of Pr i o n s a r e i n a s i n g l e t ground s t a t e . S p e c i f i c h e a t measurements o n Pr 21Ag79 [ 4 ] and d i l u t e PrxLa80-x A U * ~ a l l o y s [ 3 , 51 down t o 0.3K i n d i c a t e a predomi- nance o f s i n g l e t ground s t a t e and t h e e x i s t e n c e of v e r y low " c r y s t a l - f i e l d ' ' e x c i t a t i o n e n e r g i e s .

On t h e o t h e r hand, t h e s u p e r c o n d u c t i n g tempera- t u r e of Pr La 8 0 - x A ~ 2 0 remains r o u g h l y c o n s t a n t up t o x

1 and t h e n d e c r e a s e s f o r l a r g e r x v a l u e s .

and t h e p o i n t c h a r g e model on computer-generated c l u s t e r s of d e n s e random packing o f h a r d s p h e r e s , i n o r d e r t o c a l c u l a t e t h e e n e r g y l e v e l s o f t h e r a r e e a r t h (RE) i o n s . I t i s found t h a t t h e d i s t r i b u t i o n o f A and of t h e o v e r a l l s p l i t t i n g h a s a l m o s t a g a u s s i a n form. Concerning induced magnetism i n amorphous RE a l l o y s , t h e o n l y t h e o r e t i c a l t r e a t m e n t i s due t o H a r r i s e t a l . [ 7 ] who developed a n "ad hoc" s i n g l e t - s i n g l e t model w i t h i n t h e u n i a x i a l random a n i s o t r o p y model ; b u t t h e y assumed a f i x e d A v a l u e , which d i s - a g r e e s c o m p l e t e l y w i t h t h e low t e m p e r a t u r e s p e c i f i c h e a t d a t a and t h e c a l c u l a t i o n s o f Cochrane e t aL [ 6 ] . I1 - THE THEORETICAL MODEL.

I n t h e p r e s e n t work, we p r o p o s e a s i n g l e t - s i n - g l e t model w i t h t h e energy s p l i t t i n g A v a r y i n g ran- domly o v e r t h e RE s i t e s a c c o r d i n g t o a g a u s s i a n d i s t r i b u t i o n . D e t a i l e d c a l c u l a t i o n s w i l l b e presen- t e d i n a l o n g e r paper [ 8 ] and we j u s t g i v e h e r e t h e main r e s u l t s .

The H a m i l o t n i a n of t h e system i s g i v e n by : ( 2 ) T h i s s u g g e s t s a n induced moment t y p e b e h a v i o u r cha-

where t h e f i r s t term i s t h e CF p a r t , t h e second t h e r a c t e r i s t i c o f s i n g l e t ground s t a t e s y s t e m s , irnplyi~g

H e i s e n b e r g H a m i l t o n i a n and t h e l a s t term c o r r e s p o r d s a t h r e s h o l d exchange s t r e n g t h f o r t h e o n s e t of ma-

t o a n a p p l i e d f i e l d H. The CF p a r t d e s c r i b e s f o r g n e t i s m . Such a c o n c l u s i o n i s a l s o drawn by t h e nu-

each s i t e two s i n g l e t s (denoted by 10.2 and I I .>) c l e a r h y p e r f i n e s p e c i f i c h e a t c o n t r i b u t i o n which i s

s e p a r a t e d by a n e n e r g y s p l i t t i n g A. which i s assu- i m p o r t a n t f o r x=10 and d i s a p p e a r s f o r x=l [5]. med t o v a r y randomly o v e r t h e RE s i t e s a c c o r d i n e t o

On t h e t h e o r e t i c a l s i d e , t h e e x i s t e n c e o f a .,

a g a u s s i a n d i s t r i b u t i o n : s i n g l e t ground s t a t e f o r P r i s e a s i l y u n d e r s t o o d i n

1 ( A - ^{a o ) *}

amorphous systems a s i n c r y s t a l l i n e ones. The most P(A)

- ^exP f - 1 ^{( 3 )}

i m p o r t a n t p o i n t seems t o b e t h a t t h e energy A of =A] 2 A;

e x c i t a t i o n t o t h e f i r s t e x c i t e d s t a t e v a r i e s from Our r e s u l t s w i l l b e p r e s e n t e d a s a f u n c t i o n o f s i t e t o s i t e . Cochrane e t a l . C61 used t h e most ge- t h e parameter y = A,/% and i n t h e f o l l o w i n g we n e r a l form o f t h e second o r d e r c r y s t a l f i e l d (CF) w i l l r e p l a c e t h e p o l a r i z a t i o n matrix-element H a m i l t o n i a n ( f o r t h e ith s i t e ) : ai = < Oi I ^Jiz 1 ^{1. >} by t h e i r a v e r a g e v a l u e a .

P h y s i c a l l y , a g i v e s t h e s a t u r a t i o n moment a t 0 K.

(1) The "CF s u s c e p t i b i l i t y " ( z e r o exchange) i s g i -

+ ~ a b o r a t o i r e a s s o c i g a u C.N.R.S. (L.A. n02)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19808157

ven by t h e average of t h e Van Vleck s u s c e p t i b i l i t y : 2 g 2 4 a 2 A.

( 4 )

where FT) means an average of F(A) over t h e gaus- s i a n d i s t r i b u t i o n P(A) .

I n presence of exchange i n t e r a c t i o n , t h e para- magnetic s u s c e p t i b i l i t y x i s g i v e n i n t h e molecular f i e l d approximation 191 by :

2 2 2

2g VB" 1 2&; a2

- ^{- = -} - ¹ - A. (5)

'CF where

c h a r a c t e r i z e s t h e r e l a t i v e s t r e n g t h of t h e exchange f i e l d with r e s p e c t t o t h e CF s p l i t t i n g and h e r e '$0) i s e q u a l t o 1 tij.

j - -

The induced magnetic moment gp J=gpB <JZ>, B

which i s computed h e r e by assuming a ferromagnetic exchange i n t e r a c t i o n , i s g i v e n by t h e s e l f - c o n s i s - t e n t e q u a t i o n [9 1 ^:

-

- J

s i n e tanh [& ^{(coo 8+} s i n e )

a ] ⁽⁷⁾

w i t h :

We c a n t h e n c a l c u l a t e T / a a t 0 K a s a function of A i n o r d e r t o o b t a i n t h e t h r e s h o l d v a l u e of A.

f o r t h e o n s e t of induced magnetism. The o r d e r i n g temperature T i s t h e n g i v e n by :

- = - A

(9) F i n a l l y , t h e s p e c i f i c h e a t C v per mole of t h e RE component i s given i n t h e molecular f i e l d appro- ximation by :

where :

D/LT e F~ = ( B / ~ T ) D/kT

( I + e l 2

s i n e 1 - ( e +ZD/kT)-

F2= -

1 ^{(I + e D/kT ) 2}

Here :

A A -

J s i n e )

D = A (

a

( 1 3 )

denotes s i n g l e t - s i n g l e t s p l i t t i n g i n t h e presence of a molecular f i e l d .

I11 - RESULTS AND DISCUSSION

We p r e s e n t some r e s u l t s of t h e t h e o r e t i c a l mo- d e l i n f i g u r e s 1 t o 4 . F i g u r e 1 shows t h e reduced i n v e r s e s u s c e p t i b i l i t y l/xCF v e r s u s ~ T / A ~ f o r seve- r a l v a l u e s of y

A / A Curve A ( f o r y

114) de-

1

v i a t e s only s l i g h t l y from t h e Van Vleck behaviour of a c r y s t a l l i n e system g i v e n by t h e d o t t e d c u r v e (y

0 ) . On t h e c o n t r a r y , curve D (y

2) does n o t show any reminescence of t h e Van Vleck b e h a v i o u r , w h i l e curves B (y

112) and C ('y

1) have a n in- t e r m e d i a t e behaviour. We f i n d t h a t c u r v e C i s very s i m i l a r t o t h e experimental 1 / x v e r s u s T curves f o r d i l u t e Pr La 8 0 - x A ~ 2 0 a l l o y s ; i n p a r t i c u l a r , t h e low t e m p e r a t u r e s l o p e of c u r v e C (between

kT/Ao

0.025 and 0.05) i s about 2 times t h e h i g h t e m p e r a t u r e s l o p e .

2 2 2

2g1.1Ba

Fig. 1 : P l o t of --- -

0

XCF

c r y s t a l l i n e Y = 0 c a s e ( d o t t e d c u r v e ) , y = 114 (curve A), Y = 112 (B), y

1 (C) and y = 2 ( D ) .

F i g u r e 2 shows t h e reduced z e r o - f i e l d induced

magnetic moment JIa a t 0 K a s a f u n c t i o n of t h e

r e l a t i v e exchange s t r e n g t h A f o r d i f f e r e n t v a l u e s

of y . Curve A (y

114) i s s i m i l a r t o t h e c r y s t a l l i -

ne d o t t e d curve (y

0) with both a t h r e s h o l d v a l u e

A. c l o s e t o 1 and a s h a r p r i s e above t h a t v a l u e .

The o t h e r curves have a d i f f e r e n t behaviour w i t h

both a much s m a l l e r t h r e s h o l d v a l u e and a much slow-

l i e r r i s e above i t . Experimentally, t h e e x i s t e n c e

of an exchange t h r e s h o l d i s evidenced by t h e concen-

t r a t i o n dependence of both t h e superconducting t r a n -

s i t i o n t e m p e r a t u r e 121 and t h e o n s e t of a n u c l e a r

h y p e r f i n e s p e c i f i c h e a t c o n t r i b u t i o n [ 5 ] i n

JOURNAL DE PHYSIQUE

Fig. 2 : P l o t of t h e reduced z e r o - f i e l d induced momentYla a t 0 K v e r s u s A. f o r t h e c r y s - t a l l i n e y

0 c a s e ( d o t t e d c u r v e ) , y = 114

(curve A), y = 1 / 2 ( B ) , y = 1 (C) and y = 2 (D).

PrxLa80-xA~20 a l l o y s .

F i g u r e 3 shows t h e reduced s p e c i f i c h e a t C v /R per mole of t h e RE i o n v e r s u s kT/fio i n absence of any induced magnc :ism. The d o t t e d curve (y = 0) i s simply t h e Schottky s p e c i f i c h e a t . When y i n c r e a s e s

from 0 t o 1 , t h e low temperature d e c r e a s e of C v

changes from e x p o n e n t i a l t o almost l i n e a r . The expe- r i m e n t a l d a t a [ 5 ] on P r La

X

80-xA~20 seem t o be c l o s e r t o t h e y

1/2 curve which h a s s t i l l a p o s i t i v e cur- v a t u r e a t low temperatures t h a n t o t h e y = l o n e ;howe- v e r , t h e experimental o b s e r v a t i o n of t h e r e l a t i v e l y rounded maximum i s obscured by t h e c o n t r i b u t i o n from t h e superconducting temperature.

A C v -

l a ) # = 0.5

___--- - - _{-- --.}

.- -- --- _{--- --__}

>

( b ) 1-1

_---______ --__ _{---___}

-.

0.25 0.50 0.75 kT/Ao

F i g . 4 : P l o t of t h e s p e c i f i c h e a t C /R v e r s u s

v

kT/Ao f o r a ) y = 0.5 and b) y = 1 and f o r A = 0.75 and A.

1 . The d o t t e d curves correspond t o a z e r o magnetic moment 7

0.

F i g u r e 4 shows C v /R v e r s u s kT/A i n p r e s e n c e of induced magnetic o r d e r i n g . I n a l l t h e c a s e s c o n s i d e r e d , t h e r e i s a s h a r p peak j u s t below T c' r e f l e c t i n g t h e s h a r p d e c r e a s e of t h e induced moment a s a f u n c t i o n of temperature n e a r Tc. However, no s h a r p peak h a s been e x p e r i m e n t a l l y observed, b u t any way i t i s c e r t a i n l y n o t e a s y t o s e p a r a t e i t from t h e n u c l e a r c o n t r i b u t i o n . N e v e r t h e l e s s , according t o Garoche e t a l . 151, a s temperature d e c r e a s e s below 1 K , t h e C v curve f o r PrIOLa70A~20 r i s e s above t h a t f o r P r l L a 7 9 A ~ 2 0 and goes through a n i n f l e c t i o n p o i n t around 0.5 K, which corresponds t o a broad maximum Fig. 3 : P l o t of t h e s p e c i f i c h e a t CV/R v e r s u s of t h e d i f f e r e n c e between t h e two curves. This i s

- k ~ , A in absence of induced magnetism f o r the crys- Perhaps a reminescence of t h e peak near Tc ; t h e

0

t a l l i n e y

0 c a s e ( d o t t e d c u r v e ) , y = 1/4 (curve c a s e y = 0 . 5 and A.

0.75 i n f i g u r e 4 appears t o

A ) , y

112 ( B ) , y = 1 (C) and y = 2 (D) b e t h e most a p p r o p r i a t e f o r a comparison. On t h e

o t h e r hand, t h e low temperature t a i l of t h e two

curves corresponding t o A

1 shows a n important

d e c r e a s e w i t h r e s p e c t t o t h e curves i n t h e absence REFERENCES of magnetic o r d e r i n g ; t h i s a r i s e s from t h e increa-

1 ) C. Pappa, These 3Pme c y c l e , U n i v e r s i t s P a r i s V I s e of t h e s i n g l e t - s i n g l e t s p l i t t i n g i n t h e molecu-

( 1 . - - 979)

^-, l a r f i e l d . Such an e f f e c t can b e a l s o induced by an

2) N. Hassanain, A. Berrada, J . Durand and B. Loegel a p p l i e d f i e l d , a s observed by Garoche e t a l . [5].

J o u r n a l of Mag. and Magn. Mat. =,I377 (1980) I V - CONCLUDING REXARKS

We have developed a s i n g l e t - s i n g l e t model f o r an amorphous system by assuming a g a u s s i a n d i s t r i - b u t i o n of t h e energy s p l i t t i n g . However, we have assumed a ferromagnetic exchange i n t e r a c t i o n , which i s d i f f i c u l t t o j u s t i f y f o r amorphous a l l o y s l i k e PrxAglOO-x o r PrxLa80-xA~20 ; i n such systems, sus- c e p t i b i l i t y d a t a show t h a t t h e exchange i n t e r a c t i o n i s predominantly a n t i f e r r o m a g n e t i c , so t h a t t h e induced o r d e r i n g would perhaps b e s p i n - g l a s s - l i k e , a s i n t h e c o n c e n t r a t e d amorphous a l l o y Pr80A~20C101 Secondly, t h e assumption of an uniform molecular f i e l d seems t o b e a n o v e r s i m p l i f i c a t i o n i n a system w i t h a l a r g e d i s t r i b u t i o n of A. Thus, we a r e now s t u d y i n g an e x t e n s i o n of t h e p r e s e n t model t o one with a g a u s s i a n d i s t r i b u t i o n of t h e molecular f i e l d . Such a g a u s s i a n c e n t e r e d a t z e r o could probably b e t t e r d e s c r i b e t h e induced s p i n - g l a s s - l i k e orderixg expected i n t h e Pr a l l o y s .

In c o n c l u s i o n , i n s p i t e of t h e receding theo- r e t i c a l d i f f i c u l t i e s , t h e agreement w i t h experimen- t a l d a t a i n Pr amorphous a l l o y s i s r a t h e r s a t i s f a c - t o r y , concerning t h e shape of t h e magnetic

s u s c e p t i b i l i t y , t h e behaviour of t h e s p e c i f i c h e a t a t low temperatures o r f i n a l l y t h e t h r e s h o l d f o r t h e o n s e t of induced magnetism a s evidenced by superconducting temperature o r n u c l e a r s p e c i f i c h e a t measurements,

3) A . F e r t , , P. Garoche, B. Boucher and J . Durand, I n t . Conf. on C r y s t a l l i n e e l e c t r i c f i e l d and s t r u c t u r a l e f f e c t s i n f - e l e c t r o n systems, P h i l a d e l p h i a , November 1979

4) P. Garoche, A . F e r t , J . J. Veyssid and B.

Boucher, J. M. M. M. 15-18, 1397 (1980) 5) P. Garoche, J . J . Veyssid and J . Durand, t o b e

p u b l i s h e d

6 ) R. W. Cochrane, R . H a r r i s , M. P l i s c h k e , D. Zobin and M. J . Zuckermann, J . Phys. F. : Y e t a l Phys.

5 , 763 (1975) -

7) R. H a r r i s , ?I. P l i s c h k e and M. J. Zuckermann, Phys. Rev. L e t t . 31, 160 (1973)

8) A . K. B h a t t a c h a r j e e and B. Coqblin, t o be p u b l i - shed

9 ) B. R. Cooper i n l k a g n e t i c P r o p e r t i e s of Rare- E a r t h Metals", ed. R. J . E l l i o t , Plenum P r e s s London and New York, 1972), pp. 44-45

10) A . Berrada, J. Durand, N . Hassanain and

B. Loegel, X I V Rare-Earth Research Conf. (Fargo, North Dakota, June 1979).

We wish t o thank P. Garoche and J. J. Veyssid

f o r c o m u n i c a t i n g t h e i r s p e c i f i c h e a t d a t a p r i o r t o

p u b l i c a t i o n .