transport properties of ferromagnets

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chapter

TRANSPORT PROPERTIES OF

F E R R OM A GN E T S

I A C A M P B E L L A N D A F E R T

Laboratoire de Physique des Sofides

Universit6 Paris-Su d 91 405 Orsa y

France

Ferromagnetic Materials Vol. 3

Edited by E.P. Wohlfarth

O North-HollandPublishing Company 1982

7 7



ONTENTS

I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . .

1. G e n e r a l p r o p e r t i e s o f t r a n s p o r t in f e r r o m a g n e t s . . . . . . . . . . . . .

1 .1 . R e s i s t i v i t y a n d H a l l e f f e c t o f a m on od o m a i n p o l y c r y s t a l . . . . . . . . .

1 .1 .1 . S p o n t a n e o u s r e s i s t i v i t y a n i s o t r o p y . . . . . . . . . . . . . .

1 .1 .2 . E x t r a o r d i n a r y H a l l e f f e c t . . . . . . . . . . . . . . . . .

1 .1 3 . P l a n a r H a l l e f f e c t . . . . . . . . . . . . . . . . . . . .

1 .2 . Re s i s t i v i t y a n d H al l e f f e c t in s i n g l e c r y s t a l f e r r o ma g ne t s . . . . . . . . .

1 .3 . T h e r m a l a n d t h e r m o e l e c t r i c e f f e c t s i n p o l y c r y s t a l s . . . . . . . . . . .

2 . E l e c t r i c a l r e s i s t i v i t y o f f e r r o m a g n e t s . . . . . . . . . . . . . . . . .

2 .1 . T h e o r e t i c a l m o d e l s . . . . . . . . . . . . . . . . . . . . .

2 .1 .1 . S p i n d i s o r d e r s c a t t e r i n g . . . . . . . . . . . . . . . . .

2 . 1 .2 . T w o c u r r e n t m o d e l . . . . . . . . . . . . . . . . . . .

2 .2 . R e s i s t i v i t y o f p u r e m e t a l s . . . . . . . . . . . . . . . . . . .

2 .2 .1 . T a b u l a r r e s u l t s . . . . . . . . . . . . . . . . . . . .

2 . 2 2 . R e s i s t i v i t y a t l o w t e m p e r a t u r e s . . . . . . . . . . . . . . .

2 . 2 .3 . R e s i d u a l r e s i s t i v i t y . . . . . . . . . . . . . . . . . . .

2 .2 .4 . H i g h f i e ld b e h a v i o u r . . . . . . . . . . . . . . . . . . .

2 .3 . A l lo y s: r e si d ua l r es is ti vi ty a n d t e m p e ra t u re d e p e n d e n c e of r es is ti vi ty . . . .

2 .3 .1 . N i c k e l h o s t . . . . . . . . . . . . . . . . . . . . . .

2 3 .2 . C o b a l t h o s t . . . . . . . . . . . . . . . . . . . . . .

2 . 3 .3 . I r o n h o s t . . . . . . . . . . . . . . . . . . . . . . .

2 .3 4 . A l l o y s c o n t a i n i n g i n t e r s t i t i a l i m p u r i t i e s . . . . . . . . . . . . . .

2 .4 . H i g h t e m p e r a t u r e a n d c r i t i c a l p o i n t b e h a v i o u r . . . . . . . . . . . .

3 O t h e r t r a n s p o r t p r o p e r t i e s o f N i C o F e a n d t h e i r all~oys . . . . . . . . . .

3 .1 . O r d i n a r y m a g n e t o r e s i s t a n c e . . . . . . . . . . . . . . . . . .

3 .2 . O r d i n a r y H a l l c o e f f i c i e n t . . . . . . . . . . . . . . . . . . .

3 .3 . S p o n t a n e o u s r e s i s t i v i t y a n i s o t r o p y . . . . . . . . . . . . . . . .

3 .4 . E x t r a o r d i n a r y H a l l e f f e c t . . . . . . . . . . . . . . . . . . .

3 .5 . T h e r m o e l e c t r i c p o w e r . . . . . . . . . . . . . . . . . . . .

3 .6 . N e r n s t - E t t i n g s h a u s e n e f f e c t . . . . . . . . . . . . . . . . . .

3 .7 . T h e r m a l c o n d u c t i v i t y . . . . . . . . . . . . . . . . . . . . .

4 . D i l u t e f e r r o m a g n e t i c a l l o y s . . . . . . . . . . . . . . . . . . . .

4 .1 . P a l l a d i u m b a s e d a l l o y s . . . . . . . . . . . . . . . . . . . .

4 .1 .1 . R e s i s t i v i t y a n d i s o t r o p i c m a g n e t o r e s i s t a n c e . . . . . . . . . . .

4 . 1 . 2 . M a g n e t o r e s i s t a n c e a n i s o t r o p y . . . . . . . . . . . . . . .

4 .1 .3 . E x t r a o r d i n a r y H a l l e f f e c t . . . . . . . . . . . . . . . . .

4 . 1 .4 . T h e r m o e l e c t r i c p o w e r . . . . . . . . . . . . . . . . . . .

4 .2 . P l a t i n u m b a s e d a l l o y s . . . . . . . . . . . . . . . . . . . .

7 5 1

7 5 1

7 5 1

7 5 2

7 5 4

7 5 4

7 5 5

7 5 6

7 5 7

7 5 7

7 5 7

7 5 8

7 6 2

7 6 2

7 6 2

7 6 4

7 6 5

7 6 6

7 6 8

7 7 1

7 7 1

7 7 2

7 7 3

7 7 6

7 7 6

7 7 8

7 7 9

7 8 3

7 9 0

7 9 2

7 9 2

7 9 3

7 9 3

7 9 3

7 9 4

7 9 4

7 9 4

7 9 5

7 4 8



5 A m o r p h o u s a ll oy s

5 1 R e s i st i v it y o f a m o r p h o u s a l l o y s

5 2 H a l l e f fe c t a n d re s i s ti v i t y a n i s o t r o p y o f a m o r p h o u s a l l o y s

R e f e r e n c e s

7 9 5

7 9 5

8 0 0

8 0 0

7 4 9



Introduction

T h e s c o p e o f t h i s c h a p t e r w i l l i n f a c t b e r a t h e r m o r e r e s t r i c t e d t h a n i t s t i t l e m i g h t

s u g g e s t . W e wi l l o u t l i n e s o m e v e r y g e n e r a l p r o p e r t i e s o f t r a n s p o r t i n f e r r o -

m a g n e t s a n d w i l l s u m m a r i z e t h e m o d e l s t h a t h a v e b e e n u s e d . W e w i l l t h e n

r e v i e w i n d e t a i l r e s u l t s o n p a r t i c u l a r s y s t e m s . W e wi l l n o t t r e a t m a g n e t i c s e m i -

c o n d u c t o r s , a n d we o n l y o c c a s i o n a l l y m e n t i o n r a r e e a r t h m e t a l s a n d t h e i r a ll o y s ,

a s t h e y w i ll b e t r e a t e d i n a n o t h e r c h a p t e r * . B e c a u s e o f t h e l a c k o f s y s t e m a t i c d a t a ,

we w il l n o t r e v i e w t h e p r o p e r t i e s o f m a g n e t i c i n t e r m e t a l l i c c o m p o u n d s a s s u c h,

b u t w i l l m e n t i o n r e s u l t s o n t h e s e c o m p o u n d s a s a n d wh e n t h e y e x i s t . A s e c t i o n i s

d e v o t e d t o a m o r p h o u s f e r r o m a g n e t s . H o w e v e r , w e w il l b e m o s t l y c o n c e r n e d w i t h

t h e t r a n s p o r t p r o p e r t i e s o f th e t r a n s i ti o n f e r r o m a g n e t s F e , C o a n d N i a n d t h e i r

a l l o y s , wh i l e d r a wi n g e x a m p l e s f r o m o t h e r c l a s s e s o f m a g n e t i c m e t a l s t o i l l u s t r a t e

p a r t i c u l a r t y p e s o f b e h a v i o u r .

T h e s u b j e c t h a s a d v a n c e d s o m u c h o v e r t h e l a st 25 y e a rs t h a t t h is c h a p t e r h a s

l it tl e to d o w i t h t h e e q u i v a l e n t o n e i n B o z o r t h s b o o k . H o w e v e r , a n u m b e r o f

b o o k s a n d r e v i e w a r t i c le s h a v e b e e n v e r y u s e f u l; a l is t o f t h e m is g i v e n a t t h e e n d

of the cha p te r ( Jan 1957, M ot t 1964, H ur d 1974 and 1972 , Do r le i jn 1976).

1 Gene ral properties of transport in ferromagn ets

W e wi l l d i s cu s s e f fe c t s wh i c h a r i se q u i t e g e n e r a l l y a s a c o n s e q u e n c e o f t h e

s y m m e t r y p r o p e r t i e s o f t h e f e r r o m a g n e t i c s t a t e .

1 1 Resist iv i ty and Hal l e f fect of a mo no dom ain polycrystal

T h e c o m p o n e n t s E i o f t h e e l e c t r i c f i e l d i n s i d e a c o n d u c t o r a r e r e l a t e d t o t h e

c u r r e n t d e n s i t y ~ t h r o u g h

J~i ~ E PiJJ] , ( 1 )

J

Vol. 1, oh. 3, by Legvold.

751



752 I .A. CAMPBELL AN D A. FERT

w h e r e t h e p ij c o e f fi c ie n t s f o r m t h e r e s i st i v it y t e n s o r . S u p p o s e w e h a v e a r a n d o m

p o l y c r y s t a l w i t h it s m a g n e t i z a t i o n s a t u r a t e d in t h e d i r e c t i o n z . F r o m s y m m e t r y

a r g u m e n t s ( B ir ss 1 9 64 , H u r d 1 97 4) o n e f i n d s t h a t s u c h a m a g n e t i z e d i s o t r o p i c

m e d i u m h a s a re s i st iv i t y t e n s o r o f t h e f o r m :

[_ orP± B) p H B ) 0 1

[P~i] = | p H ( B ) p a B ) 0 .

0 RjI(B)

(2)

T h i s f o r m o f t h e r e s i s ti v it y t e n s o r c o r r e s p o n d s t o t h e f o l lo w i n g e x p r e s s i o n o f t h e

e l e c t r i c f i e l d E :

E = p I ( B ) J + [ P l I ( B ) - p ± (B )] [o ~ • J ] ~ +

p H B ) o z x J ,

3 )

w h e r e J i s t h e c u r r e n t v e c t o r a n d a is a u n i t v e c t o r in t h e m a g n e t i z a t i o n d i r e c ti o n .

T h e p i i B ) a r e f u n c t i o n s o f t h e i n d u c t i o n B , w h i c h d e p e n d s o n t h e e x t e r n a l f ie ld H

a n d o n t h e d e m a g n e t i z i n g f a c t o r D o f t h e p a r t ic u l a r s a m p l e g e o m e t r y ,

B = H + 4 r r M ( 1 - D ) . 4 )

T h i s i s b e c a u s e o f L o r e n t z f o r c e e f f e c ts w h i c h e x i s t in a n y c o n d u c t o r . T h a t t h e

e f f e c ti v e f ie ld a c t i n g o n t h e e l e c t r o n t r a j e c t o r i e s i n a f e r r o m a g n e t i s i n d e e d B

s e e m s p h y s ic a l ly r e a s o n a b l e ( K i t te l 1 9 63 ) a n d h a s b e e n v e r i f i e d e x p e r i m e n t a l l y

( A n d e r s o n a n d G o l d 1 9 6 3 , T s u i 1 9 6 7 , H o d g e s e t a l . 1 9 6 7 ) . C o n v e n t i o n a l l y , t h e

c o e f f i c i e n t s P i j a r e s p l i t u p i n t o t w o p a r t s

p i j B ) = p ij + p° B )

w h e r e t h e p~j a r e t h e

s p o n t a n e o u s o r e x t r a o r d i n a r y c o ef fi ci en t s, a n d t h e

p ° B )

a r e t h e o r d i n a r y

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

s o m e f o r m o f e x t r a p o l a t io n b e c a u s e o f th e p r e s e n c e o f th e i n t e r n al L o r e n t z f ie ld .

I n p r a c t i c e , t h i s e x t r a p o l a t i o n c a u s e s l i t tl e d if f ic u l ty e x c e p t f o r f a ir l y p u r e s a m p l e s

a t l o w t e m p e r a t u r e s w h e r e B / p c a n b e h i g h a n d t h e o r d i n a r y e f f e c t s b e c o m e l a rg e .

A s s u m i n g t h a t t h is e x t ra p o l a t i o n t o z e r o B h a s b e e n m a d e w e h a v e t h e t h re e

s p o n t a n e o u s p a r a m e t e r s :

Pll = t he r e s i s t i v i t y f o r J pa r a l l e l t o M a t B = 0 ,

p ± = t h e r e s is t i v i ty f o r J p e r p e n d i c u l a r t o M a t B = 0 ,

P H = t h e e x t r a o r d i n a r y H a l l r e s i s ti v i ty .

1.1.1. Spontaneous res is t iv i ty anisotropy

T h e f a c t t h a t t h e d i a g o n a l e l e m e n t s Pll a n d p ± i n (2 ) a r e u n e q u a l m e a n s t h a t t h e

r e s is t iv i ty d e p e n d s o n t h e r e l a t iv e o r i e n t a t i o n o f M a n d d . T a k i n g t h e g e o m e t r y o f

f ig . 1 a n d c a l li n g 0 t h e a n g l e b e t w e e n M a n d J , a s b y d e f i n i t io n t h e r e s i s t iv i t y i s

p = E . J / f J I 2

w e h a v e f r o m e q . ( 3 ) ,



T R A N S P O R T P R O P E R T I E S O F F E R R O M A G N E T S 7 5 3

[

7

F i g. 1 . E x p e r i m e n t a l g e o m e t r y f o r t r a n s p o r t p r o p e r t i e s . T h e c u r r e n t J i s c o n s t r a i n e d t o fl o w a l o n g t h e

d i r e c t i o n x , a n d w h a t e v e r t h e a p p l i e d f i el d o r m a g n e t i z a t i o n d i r e c t i o n , t h e r e s i st i v it y i s p r o p o r t i o n a l t o

t h e v o lt a g e b e t w e e n p r o b e s A a n d

B, p = V B - VA)b t /J1 .

F o r t h e c o n v e n t i o n a l H a ll g e o m e t r y , t h e

a p p l i e d f ie l d o r m a g n e t i z a t i o n d i r e c t i o n i s a l o n g z a n d t h e H a l l v o l t a g e is m e a s u r e d b e t w e e n C a n d D .

T h e n P H

= VO-

V c) t / J

a n d R 0 = V D -

V c ) t / J H

o r R ~ = V D -

Vc ) t / J 4~rM.

P ll + 2 p ~ + c o s 2 0 - ½ ) P l l - P ~ ) 5 )

p s = 0 = 3

T h e r e l a ti v e s p o n t a n e o u s a n i s o t r o p y o f t h e r e s is t iv i ty i s d e fi n e d a s:

A p _ P l I - P ±

P 3Pl l 2p±

6 )

I t c a n h a v e e i t h e r s i g n , a n d w h i l e g e n e r a l l y v a l u e s o f a f e w p e r c e n t a r e ty p i c a l, c e r t a i n

s y s t em s s h o w m o r e t h a n 2 0 a n i s o tr o p y .

F i g u r e 2 s h o w s s c h e m a t i c a l l y t y p i c a l r e s i s ti v i ty c h a n g e s a s a f u n c t i o n o f a p p l i e d

f ie l d , a n d t h e m e t h o d o f e x t r a p o l a t i o n t o o b t a i n Pll a n d p ± i s i n d i c a t e d . N o t e t h a t

t h e r e s is t iv i ty o f t h e z e r o f i e l d s t a te d e p e n d s o n t h e e x a c t d o m a i n c o n f i g u r a t i o n , s o

i t i s h i s t o r y d e p e n d e n t a n d i s n o t w e l l d e f in e d e v e n f o r a g i v e n s a m p l e a t a g i v e n

t e m p e r a t u r e . I n t h e s a m e w a y , t h e c h a n g e o f r e si s ti v it y w i t h f i el d b e l o w t e c h n i c a l

{ 8 =0 } . . . .

~a=01-- ,

H

I

B/p 8 = 0 1

F i g. 2. S c h e m a t i c e x t r a p o l a t i o n f o r a f e r r o m a g n e t i c o f

p H )

r e s is t iv i ty c u r v e s t o B = 0 . T h e h e a v y

l i n e s i n d i c a t e o b s e r v e d r e s i s t i v i t y a s a f u n c t i o n o f f i e l d w h e n t h e f i e l d i s a p p l i e d p a r a l l e l

1 1 )

o r

p e r p e n d i c u l a r ± ) t o t h e c u r r e n t d i r ec t io n . A r r o w s s h o w t h e r e g i o n s o f i n c o m p l e t e t e c h n i c a l s a t u r a ti o n .

D o t t e d l i ne s i n d ic a t e e x t r a p o l a t i o n s f r o m t h e s a t u r a t e d r e g i o n s t o t h e r e s p e c t i v e B = 0 p o i n t s

B = H + 4 ~ - M 1 - D ) w h e r e D i s t h e d e m a g n e t i z i n g f a c t o r fo r t r a n s v e r s e o r l o n g i t u d i n a l f ie l ds ) .



754 I.A. CAMPBE LL AND A. FERT

sa tu r a t ion d e pe nds on the m a gne t i z a t ion p roc ess . We ha ve a s sum e d a bove tha t in

the sa tu r a t e d r e g ion on ly Lor e n tz f o r c e e f f e c t s a r e im por t a n t f o r the va r i a t ion o f

Pll or p± with B. In fact , the e xtern al f ie ld H m ay also increase th e m agn etiz at io n

of the sam ple w hich wi ll a f fec t p , because of a reduc t io n in sp in d iso rder

resist ivi ty. This is par t icular ly important near T¢.

1.1.2. Extraordinary Hall effect

The of f -d iagona l te rms +pH in 2) lead to the ext raordin ary Hal l vol tage ,

EH B = 0) = pHa × J ,

7)

perpen dicula r to M and J . This is usua l ly mea sured in the conve nt iona l Ha l l

geo me try wi th M perpe ndicu la r to J fig . 1). W e can a lso def ine the ext rao rdina ry

Hal l angle

¢hH = P~/ Pi 8)

and, by ana logy wi th th e d ef in i t ion of the o rdina ry H al l coef f ic ient R0 = p ° / B w e

have the ext raordinary Hal l coef f ic ient

R s = p n / 4 ~ M .

9)

Once aga in , ex t rapola t ion of the Hal l vol tage curve as a func t ion of B f rom the

satu rate d re gion b ack to B = 0 is nece ssary to o btain PH fig. 3). This is easy in the

low f ie ld regim e oJc~ 1 we is the cyclo tron fr equ enc y a nd ~- is the e lectro nic

r e l a xa tion t im e ) bu t t he se pa r a t ion b e twe e n o r d ina r y a nd e x t r a o r d ina r y e f fe c ts i s

d i ff icul t wh en ~o~- ~ 1 whe re the ordin ary H al l vo l tage i s no longer l inear in B .

~ ~ 0 ]

B O

H

I

Fig. 3. Schemat ic extr apol atio n for a ferro mag net of a Hall resistivity curve to B = 0. On B = 0) is the

extraordinary Hall resistivity.

1.1.3. Pla na r Ha ll effect

This is a mis nom er , a s was a l ready poin te d out by Jan 1957). W hen J i s in the

di rec t ion x , the vol tage is me asu red perp endic ula r to J in the d i rec t ion y and M is

r o ta t e d in the p l a ne

xy

the n fro m eq. 4):



TRANSPORT PROPERTIES OF FERROMAGNETS 755

Ey = P/I- P±) cos 0 sin O ,

where j r . M = cos 0 . Thi s i s a mani fes ta t ion of the res i s t iv i ty an i so t ropy and i s an

ef fec t even in f i e ld which has no th ing to do wi th the Ha l l e f fec t .

1.2. Res is t iv i ty an d Ha ll e f fec t in s ingle crys tal ferrom agne ts

T h e s ym me t r y a r g ume n t s c a n be e x t e nd e d t o s ing le cr y s ta l s s e e, e. g. , H u r d 1974) .

I t i s f ound t ha t e ve n f o r c ub i c c r y s t a l s , t he r e s i s t i v i t y be c ome s de pe nde n t on t he

o r i e n t a t i ons o f t he c u r r e n t a nd t he m a gne t i z a t i on w i t h r e s pe c t t o t he c r y s ta l a xe s ,

a nd t he e x t r a o r d i na r y H a l l e f f e c t de pe n ds on t he ma gn e t i z a t i on d i r e c t i on .

Fo r a cubic fe r rom ag ne t wi th M in the d i r ec t ion 11, 12, 13) and J in the

dir ect ion /31,/32,/33) we ha ve the D6r ing ex pre ss io n D6 ring 1938):

O Oo[l+k1 a2/321+ 2 2 2 2 1

~. ..{- 0/3 /33 -- ~)

f 2/32

+ k2 20L1o~2/31/32 + 2o l2o,3/32/ 33 + 2ce3o q/33 /31 ) + k 3 S _ 1 )

..~ 4 2

k , ~ 1 / 3 1 + ~ / 3 2 + 4 2 z

1 3 / 3 3 + 3 S - I )

+ k5 2~i~2,~2/31/3~ + 2~2,~,-,~/32/3 , + 2 - , ~ 1- ~ /3 , / 33 1,

wh er e S 2 2.~_ 2 2 2 2

OL20~3 -}- 0~30~ 1

10)

O / la 2

O t he r e qu i va l e n t de f i n it i ons ha ve be e n g i ve n . K i t t e l a nd V a n V l e c k 1960)

sugges ted tha t a phys ica l ly more s ign i f i can t de f in i t ion for the magne toe las t i c

coef f i c i en t s which hav e the same sym me t ry as the res i s t iv ity coef f i c i en ts ) wou ld

r e g r oup t he t e r ms t o g i ve ne w c oe f f i c i e n t s : K1 = k l 6k4; K2 = k2 + ~k5;I K3 =

k3 2k4; K4 = k4; / 5 = }ks and p ; = p0 1- 2k4) . Thi s w ould sep ara te ou t t e rm s

hom oge ne o us l y s e c o nd o r d e r K 1 ,/ £2 ) a nd f ou r t h o r d e r / 3 , K 4, K s ) i n t he ma g-

ne t i za t ion .

The res i s t iv i ty

p = t~ + p , - a i ) c o s 2 0 - 1)

11)

f o r a m a gne t i z e d c ub i c po l yc r ys t a l is r e l a t e d t o t he t w o t ype s o f c oe f fi c ie n t s by :

t ~ = t ~ o 1 - ~ k 3 ) - = o ;

a n d

2 3 12 3 2

Pfl- P. = po ~kl + 3k2 + ggk4 + 5~k5) ~ po ~K1 + 3/ 2).

12)

T h e r e s is t iv i t y o f a de m a gn e t i z e d c ub i c po l yc r ys t a l is

memag = P ~-- p0 1 - - ~k 3) --= p ;

13)



756 I.A. CAM PBELL AND A. FERT

if the magne t iza t ion i s randomly or ien ted wi th respec t to the c rys ta l axes . In

contrast , i f there are easy axes, Pdemag s dif ferent f rom/5, for example ,

Pdemag = /90

for 111) easy axes.

The D6r ing express ion only g ives the leading te rms in an inf in i te expans ion .

Highe r o r de r t e r m s c a n be ne c e ssar y in in t er p r e t ing e xpe r im e n ta l r e su lt s .

For the ext raordinary Hal l e f fec t in a cubic monocrys ta l , ins tead of de f in ing the

z-axis as the M di rec t ion , we use the c rys ta l axes to de f ine x , y an d z . Then the

asym metr ic pa r t of the res is tiv i ty tensor beco mes Hu rd 1974).

[ o -a 3 A2 1

A3 0 - A 1 wi th

Ai = o~i eo+ ela~ + e2c~4+ e3S).

14)

-A 2 A1 0

This g ives a Ha l l vol tage which must be pe rpendicula r to J , but i s no longer

necessa r i ly pe rpendicula r to M, and which i s an iso t ropic .

We might note tha t in cubic c rys ta ls the ord inary Hal l e f fec t i s i so t ropic in the

low f ie ld l imit .

F o r f e r r om a gne t i c m onoc r ys t a l s w i th o the r sym m e t r i e s suc h a s he xa gona l ,

anal ogo us expre ssions can be der ive d Birss 1964, H ur d 1974) .

1.3. Thermal and thermoelectric effects in polycrystaIs

The the r m a l c onduc t iv i ty a nd the R igh i - Le duc e f f e c t s a r e the pu r e the r m a l

ana logues of the res is t iv i ty and the Hal l e f fec t , wi th hea t cur rents rep lac ing

e lec t r ica l cur rents and the rmal gradients rep lac ing e lec t r ica l f ie ld gradients . The

phe nom e no log ic a l s e pa r a t ion in to spon ta ne o us a nd o r d ina r y e f fe c ts c a rr ie s ove r

ent i re ly . The only d i f fe rence l ie s in the necess i ty to de f ine whe ther the exper i -

men ta l condi t ions a re i so therm al or ad iaba t ic Jan 1957).

S im ila rly , f o r a m a gne t i z e d f e r r om a gne t i c po lyc r ys t a l sub je c t t o a t e m pe r a tu r e

gradient VT, the resultant e lectr ic f ie ld is

E = E Si V T

with

S/j = NE S± 0

S~l

Here , SII and S l a re the i so therm al the rm oelec t r ic powers in pa ra l le l and p er -

pe nd ic u la r ge om e t r i e s . SNE r e p r e se n t s the spon ta n e ous Ne r n s t - E t t ing sha us e n

effect.



TRANSPORT PROPERTIES OF FERROMAGNE TS 757

F or bo t h t he t he r m a l e f f e c t s a nd t he t he r m oe l e c t r i c e f f e c ts t he ge ne r a l i z a t i on t o

monocrys ta l s i s jus t the same as for the e lec t r i ca l e f fec t s .

2 Electrical resistivity of ferroma gnets

2.1. Theoretical models

F or non - ma gne t i c me t a l s , c u r r e n t i s c a r r i e d by e l e c t r ons w h i c h a r e s c a t t e r e d by

ph ono ns a nd by i mpur i t ie s o r de f e c t s . T o a fi rs t a pp r o x i ma t i on t he t w o s c a t t e r ings

give addi t ive cont r ibu t ions to the res i s t iv i ty

p r ) = Po + p p ( T ) ,

where p0 i s the res idua l res i s t iv i ty and pp T) i s the pure meta l re s i s t iv i ty a t

t e m pe r a t u r e T . T h i s is know n a s M a t t h i e s s e n s r u l e . D e v i a t i ons f r o m t h i s r u l e a r e

s ma ll a n d a r e ge ne r a l l y e xp l a i ne d i n t e r ms o f d i f f e r e nc e s i n the a n i s o t r op i e s o f t he

r e l a xa t i on t i me o f i mpur i t y a nd o f phonon s c a t t e r i ng ove r t he F e r mi s u r f a c e

(Du gda le e t a l. 1967). In mag ne t i c m eta l s a num be r of new e f fec t s app ear .

2.1.1. Spin disorder scattering

O n t he s i mp l e mod e l o f w e l l de f i ne d l oca l mo me n t s i n a s imp l e c ondu c t i on ba nd ,

a n e xc ha ng e in t e r a c t i on be t w e e n t he l oc a l a nd c ondu c t i on e l e c t r on s pi n s, J a nd s

r e s pe c t i ve l y , o f t he t ype Fs. J wil l g ive r i s e to sp in d i sorder sca t t e r ing (Kasuya

1956 , 1959 , D e G e nne s a nd F r i e de l 1958 , V a n P e s k i T i nbe r ge n a nd D e kke r 1963) .

W e l l a b o v e t h e o r d e r i n g t e m p e r a t u r e t h e r e w i l l b e a t e m p e r a t u r e i n d e p e n d e n t

pa r a m a gne t i c r e s is t iv i t y

- kv mF)2 J J +

1)

pM - 4~e2 zh3

15)

w h e r e

kF

i s t he F e r mi w a ve ve c t o r a nd z t he numbe r o f c onduc t i on e l e c t r ons pe r

a tom. Thi s pa ramagne t i c res i s t iv i ty i s ac tua l ly the sum of a non-sp in- f l ip t e rm (due

to

szJz

i n t e r a c t i ons a nd p r op o r t i on a l t o j 2 ) a nd a sp i n- fl ip t e r m ( due t o

s -J + + s+J

a nd p r opo r t i ona l t o J ) . F o r t he r e g i on a r oun d T c t he r e si s ti v it y de pe nds o n t he

sp in-sp in cor re la t ion func t ion ( see sec t ion 2 .4) . As T drops both the sp in- f l ip and

non-sp in- f l ip sca t t e r ing begin to f reeze out . Kasuya (1956) f inds

PM J- I< S )I ) J + 1 + KS)t)

~ m - - j j _}_ 1 )

(16)

F i na l l y , a t l ow t e mpe r a t u r e s ma gne t i c s c a t t e r i ng on l y r e ma i ns a s ma gnon- e l e c t r on

s c a t t e r i ng . A numbe r o f a u t ho r s ha ve c a l c u l a t e d t he l ow t e mpe r a t u r e f o r m o f t h i s

sca t t e r ing . I f the sp in ~ and sp in + con duc t ion e lec t ron Ferm i sur faces a re

a s s ume d t o be i de n t ic a l s phe r e s w i th t he s a m e r e l a xa t i on r a t e s , t he c on t r i bu t i on t o



758 I .A. CAM PBELL AN D A. FERT

t h e r e s i s t i v i t y i s p r o p o r t i o n a l t o

T

( V o n s o v s k i i 1 9 4 8 , T u r o v 1 9 5 5 , K a s u y a 1 9 5 9 ,

M a n n a r i 1 9 59 ). M a n n a r i ( 19 5 9 ) f in d s *

_ ¢r3 N J m F 2 ( _ ~ ) 2

Pm 32

e Z z h E ~

( k T ) 2

1 7 )

w h e r e / x e i s t h e e f f e c t i v e m a s s o f t h e m a g n o n s (/X e= h 2 / 2 D ) . T h e r e s is t iv i t y v a r ie s

a s T 2 b e c a u s e t h e l o ss o f m o m e n t u m o f t h e t o t a l e l e c t r o n s y s t e m d u e t o c o l li si o ns

w i t h t h e m a g n o n s i s p r o p o r t i o n a l t o

q 2 d q

e x p ( D q ~ / k T ) - 1

w h i c h i s e q u a l t o a c o n s t a n t t i m e s T 2. H e r e q i s t h e m a g n o n w a v e v e c t o r . T h e

f i n a l q 2 f a c t o r i n s i d e t h e i n t e g r a l i s a s m a l l a n g l e s c a t t e r i n g f a c t o r , a n d t h e r e s t o f

t h e in t e g r a n d r e p r e s e n t s th e n u m b e r o f m a g n o n s w i t h v e c t o r q at t e m p e r a t u r e T

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

t h e w e l l k n o w n T 5 l im i ti n g d e p e n d e n c e f o r e l e c t r o n - p h o n o n s c a tt e ri n g i n n o n -

m a g n e t i c m e t a l s e x c e p t t h a t:

( i) t h e d i s p e r s i o n r e l a t i o n f o r m a g n o n s i s E =

D q 2

i n s t e a d o f E =

a q

f o r

p h o n o n s .

( ii) t h e c o u p l i n g s tr e n g t h f o r e l e c t r o n - m a g n o n c o l li s io n s is i n d e p e n d e n t o f q

i n s t e a d o f p r o p o r t i o n a l t o q a s i t i s f o r e l e c t r o n - p h o n o n c o l li s io n s .

I f t h e c o n d u c t i o n b a n d is p o l a r i z e d o r i f t h e r e i s s c a t t e r in g f r o m s t o d F e r m i

s u r fa c e s o r i f t h e r e is a m a g n o n e n e r g y g a p , t h e e l e c t r o n - m a g n o n s c a t te r in g w i ll

t e n d t o d r o p o ff e x p o n e n t i a l ly a t l o w t e m p e r a t u r e i n s te a d o f t h e T 2 b e h a v i o r

( A b e l sk i i a n d T u r o v 1 96 0, G o o d i n g s 1 96 3). O t h e r m e c h a n i s m s g iv in g v a r i o u s

t e m p e r a t u r e d e p e n d e n c e s h a v e b e e n a l s o s u g g e s t e d ( V o n s o v s k i i 1 9 5 5 , T u r o v a n d

V o l o s h i n s k i i 1 9 6 7) b u t a p p e a r t o g i v e e x t r e m e l y s m a ll c o n t r i b u t i o n s ( s e e s e c t i o n

2.2) .

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

h a v e t h e s a m e r e l a x a t i o n t im e s a n d c a r r y t h e s a m e c u r r e n t . I f t h e r e a r e d i f f e re n t

s p in 1 a n d s p i n { c u r r e n t s , t h e r e w il l b e a c o n t r i b u t i o n o f t h e m a g n o n s t o t h e

r e si s ti v it y t h r o u g h s p in m i x i ng . T h i s a p p e a r s t o b e t h e m a j o r m a g n o n c o n t r i b u t i o n

i n m a n y f e r r o m a g n e t i c a l l o y s. T h i s q u e s t i o n W i ll b e t r e a t e d i n s e c t i o n 2 .1 . 2.

F i n a l l y , i n d i l u t e r a n d o m f e r r o m a g n e t s (P d_ Fe a t l o w c o n c e n t r a t i o n s is t h e b e s t

e x a m p l e o f th is ty p e o f sy s t e m ) t h e i n c o h e r e n t p a r t o f t h e m a g n o n s c a t t e r i n g -

w i th o u t m o m e n t u m c o n s e r v a t i o n - b e c o m e s p r e d o m i n a n t . T h e s c a tt er in g r a te is

t h e n s i m p l y p r o p o r t i o n a l t o t h e n u m b e r o f m a g n o n s g i v in g p ( T ) - p o - T 3 / 2

( T u r n e r a n d L o n g 1 9 7 0 , M i l l s e t a l . 1 9 7 1 ) .

2 .1 .2 . T w o c u rr e n t m o d e l

A l l t h e d i s c u s s i o n g i v e n u p t o n o w h a s n e g l e c t e d a n y s p i n i n d e p e n d e n t p o t e n t i a l .

I f t h e r e i s a s c a t te r i n g p o t e n t i a l V a t m a g n e t i c s i t e s i n a d d i t io n t o

F s • J ,

t h e n a t

* No te that the result of K asuya (1959) differs from eq. (17) by a factor

3z/ n .



TRANSPOR T PROPERTIES OF FERROM AGN ETS 759

l o w t e m p e r a t u r e s t h e s p in I' a n d s p in + e l e c t r o n s w i ll b e s u b j e c t t o p o t e n t i a l s

V + F J a n d V - F J r e s p e c t i v e l y , t o g e t h e r w i t h w e a k s p i n - f l i p s c a t t e r i n g b y

m a g n o n s ( f r o m

F ( s + J - + s - J + ) ) .

T h i s w i ll g i v e r i s e t o d i f f e r e n t s p i n t a n d s p i n $

c u r r e n t s . A l t e r n a t i v e l y , in t e r m s o f th e s - d b a n d m o d e l w h i ch i s w i d e l y u s e d f o r

t r a n s i ti o n f e r r o m a g n e t s , t h e d t a n d d + d e n s i t i e s o f s t a te s a t t h e F e r m i l e v e l a r e

d i f f e r e n t s o t h e s t o d s c a t t e r i n g r a t e s w i l l b e d i f f e r e n t f o r s p in 1' a n d s p i n $

c o n d u c t i o n e l e c t r o n s . T h i s a p p r o a c h w a s u s e d e a r l y o n t o e x p l a i n t h e

p ( T )

o f

t r a n s i t i o n f e r r o m a g n e t s u p t o a n d a b o v e t h e C u r i e t e m p e r a t u r e ( M o t t 1 9 3 6 , 1 9 6 4 ) .

I t n o w a p p e a r s t h a t t h e s im p l e s - d b a n d m o d e l a p p r o a c h i s q u e s t i o n a b l e a t h i g h

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

b a n d f e r r o m a g n e t su c h a s N i . A t l ow t e m p e r a t u r e s h o w e v e r w h e r e

Sz

i s a l m o s t

a g o o d q u a n t u m n u m b e r t h e s - d m o d e l is m o r e a p p r o p r i a t e .

T h e e l e c t r o n i c s t r u c t u r e o f t h e t r a n s it io n f e r r o m a g n e t s h a s b e e n s t u d ie d in -

t e n si v e ly , a n d l o w t e m p e r a t u r e m e a s u r e m e n t s s u ch a s t h e d e H a a s v a n A l p h e n

e f f ec t in p u r e m e t a l s a m p l e s s h o w t h a t t h e y h a v e c o m p l e x F e r m i s u r f a ce s o f m u c h

t h e s a m e t y p e a s t h o s e o f n o n - m a g n e t i c t r a n s i ti o n e l e m e n t s e x c e p t t h a t k 1' a n d

k ~ s t at e s a r e n o t e q u i v a l e n t. D e t a i l e d b a n d s t r u c t u r e m o d e l s h a v e b e e n s e t u p

w h i c h a r e i n g o o d 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 a n d w h i c h s h o w t h a t t h e

e l e c t r o n s w a v e f u n c t i o n s a r e s a n d d l i k e , g e n e r a l l y h y b r i d i z e d ( V i s s c h e r a n d

F a l i c o v 1 97 2). A n e x t r e m e s - d m o d e l w h e r e s - d h y b r i d i z a t i o n is a s s u m e d t o b e

w e a k p r o v i d e s a g o o d b a s i s f o r t h e d i s c u ss i o n o f a w i d e r a n g e o f a ll o y p r o p e r t i e s

( F r i e d e l 1 9 6 7 ) .

W e w ill n o w c o n s i d e r t h e r e s i st iv i ty a g ai n. Q u i t e g e n e r a l l y , at l o w t e m p e r a t u r e

t h e e l e c t r o n s p i n d i r e c t i o n i s w e l l d e f i n e d i f w e i g n o r e m a g n o n s c a t t e r in g a n d

s p i n - o r b i t e f f e c ts . T h e n i n a n y m o d e l w e w i ll h a v e c o n d u c t i o n i n p a r a l le l b y t w o

i n d e p e n d e n t c u r r e n t s . I f t h e c o r r e s p o n d i n g r e si s ti v it ie s a r e P t , P + th e t o t a l

o b s e r v e d r e s i s t i v i t y i s * :

= P t P +

(18)

P

t

+ P+

I n s i d e e a c h p ~ w e c a n h a v e c o m p l i c a t i o n s s u c h a s s a n d d b a n d s b u t e q . ( 1 8 ) s t i l l

r e m a i n s s tr ic t ly va li d . I f n o w t h e r e is t r a n s f e r o f m o m e n t u m b e t w e e n t h e t w o

c u r r e n t s b y s p in m i x in g s c a t t e ri n g s ( e .g . e l e c t r o n - m a g n o n o r s p i n - o r b i t s c a t te r i n g )

t h e n a g a i n q u i t e g e n e r a l l y ( F e r t a n d C a m p b e l l 1 9 7 6 ) ,

PtP+ + Pt + Pt + P+)

= , (19)

P p t + p ~ + 4 p t ~

w h e r e

P* = P * t / X ~ + P t / X t X ~ , P = P ~ / X 2 ~ + P t / X t X ~ ,

2o)

a n d

* Electrons with m agnetic mo men t parallel to the total m agnetization, i.e., electrons of the ma jority

spin band, are indicated by t ; electrons of the m inority spin ba nd by $ :



760 I .A . C A M P B E L L A N D A . F E R T

P ~ ~ = - P t + / X , X + .

H e r e p ~ , a r e i n te g r a l s o v e r t h e t ra n s i t io n r a t e s

p ( k c r , k ' c r ' )

f o r s c a t t e r i n g f r o m o n e

e l e c t r o n i c s t a t e to a n o t h e r a n d X ~ a r e i n t e g r a ls o v e r d r iv i n g t e rm s :

V k ( e O f k / O E k ) E w h e r e E is t h e e l e c t r i c f ie ld , v k t h e e l e c t r o n v e l o c i t y a n d f k t h e F e r m i

f u n c t i o n ( Z i m a n 1 9 6 0 ) .

I f t h e o c c u p a t i o n n u m b e r o f e a c h k s t a t e u n d e r z e r o e l e c tr i c fi el d is ~ , t h e n w e

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

f k = ~ 4 ) k ( d f ° k l d E k ) .

(21)

W e c a n u s e t h e v a r i a t i o n a l p r i n c i p l e , u s i n g k . u a s a t r ia l f u n c t i o n f o r t h e

f u n c t i o n k , w h e r e u i s a u n i t v e c t o r p a r a l l e l to t h e a p p l i e d f i el d . W e t h e n g e t :

f

'~ - X ~ k B T ~ , ( k . u ) [ ( k - k ' ) u ] P ( k o - , k ' o ) d k d k '

(22)

f

* ~ X t X , t k B T ( k ' • u ) ( k • u ) P ( k t k '+ ) d k d k ' ,

(23)

w h e r e P ( k o - , k ' o - ' ) i s t h e e q u i l i b r i u m s c a t t e r i n g r a t e b e t w e e n ( kc r) a n d ( k 'o - ') .

M a t t h i e s s e n s ' r u l e is a s s u m e d f o r e a c h o f t h e t h r e e t e r m s

w h e r e x r e f e r s t o d if f e r e n t t y p e s o f s c a t t e r in g c e n t r e s ( p h o n o n s , m a g n o n s , e a c h

s o r t o f i m p u r i ty ) . T h e r e a r e t e m p e r a t u r e i n d e p e n d e n t p u r e m e t a l t e r m s P it ( T ) ,

p i + ( T ) a n d p ~ ~ ( T ) w h ic h g o t o z e r o a t z e ro t e m p e r a t u r e a n d a r e a s s u m e d t o b e

i n d e p e n d e n t o f i m p u r i t y c o n c e n t r a t i o n i n d i l u te a l lo y s , a n d t h e r e a r e im p u r i t y

t e r m s w h i c h a r e a s s u m e d t o b e t e m p e r a t u r e i n d e p e n d e n t . T h e s p in - fl ip sc a t te r in g

b y i m p u r i t ie s ( v ia s p i n - o r b i t c o u p l i n g ) is g e n e r a l l y n e g l e c t e d , w h i c h l e a v e s t w o

i m p u r i t y t e r m s P0 t a n d p 0 ~ .

T h e m o d e l t h e n p r e d i c t s a d e v i a t i o n f r o m M a t t h i e s s e n s ' r u l e i n t h e r e si d u a l

r e s i s ti v i t y o f t e r n a r y a l l o y s ( F e r t a n d C a m p b e l l 1 9 76 ):

( O/A - - a B ) 2 p A P B

A p = P A B - - ( P A + P B ) - - a , , ) 2 a B p A + ( 1 + a B)Z a o B

(24)

w h e r e

a A = P A $ / P A r

a n d O /B = P B / P B

t

T h e a n a l y s is o f th e d e v i a t i o n s i n a l l o y s w i t h d i f f e re n t r e l a t i v e c o n c e n t r a t i o n s o f A

a n d B c a n b e u s e d t o d e t e r m i n e a A , a B ( C a m p b e l l e t a l. 1 96 7 , C a d e v i l l e e t a l.



TRANSPORT PROPERTIES OF FERROMAGNET S 761

1968, Dor le i jn and Miedema 1975a , Fe r t and Campbe l l 1976 , Dor le i jn 1976) .

F o r a b i na r y a l l oy a t f i n i t e t e m pe r a t u r e t he ge ne r a l e qua t i on ( 19 ) c a n be u s e d ,

wi th

P~ = Po~ + pi ~( T) .

F o r t h e l o w t e m p e r a t u r e r a n g e w h e r e

Po t, po+ >> p~t (T ), pi+ (T ), P t + T ) ,

t h i s reduces to :

(1

p ~ (T ) + k ~ + - l ] P t ( T ) ,

(25)

w h e r e

tx = Pil (T )

pi~: ( T ) '

Po = PotPo+ (res idu al resist i vity )

Pot +Po~

a n d

p i ( T ) = Pi T(T )P i ~ (T ) (which i s no t the idea l pure meta l re s i s t iv i ty ) .

p i , r (T)+ pi (T )

T h e t e r m o f e q . ( 25 ) p r op o r t i on a l t o p$ ~ ( T ) w il l g i ve a st r ong va r i a t i on o f t he

r e s is t iv i t y a s a f unc t i on o f t he t e m pe r a t u r e w he n a is ve r y d i f fe r e n t f r om un i t y ; in

n i c ke l , f o r e xa m pl e , C o , F e o r M n i m pur i t i e s e nha nc e t he l ow t e m pe r a t u r e

r e s i s t i v i t y va r i a t i on by a l m os t a n o r de r o f m a gn i t ude . T he f o r m o f t he e l e c t r on -

ma gn on con t r ibu t ion to p~ $ (T ) has b een ca lcu la ted by Fer t (1969) and Mi l ls e t a l.

( 971) us ing a sp in-sp l it sphe r i ca l con duc t ion ban d mo de l . In a l loys wi th a -~ 1 the

t e m pe r a t u r e de pe nde nc e o f t he r e s i s t i v i t y w i l l ne a r l y be t ha t o f

p i (T ) .

U s i ng e x pe r i m e n t a l d a t a on b i na r y a nd t e r na r y a ll oys , i t t u r n s ou t t o be pos s i b l e

t o ob t a i n c ons i s t e n t va lue s f o r t he pa r a m e t e r s f o r a num be r o f i m pur i t ie s , t oge t h e r

w i t h e s t i m a t e s o f t he t e m pe r a t u r e be ha v i ou r o f t he pu r e m e t a l t e r m s ( s e e s e c t i on

2 .3) . I t i s impor tan t to no te tha t i t i s no t poss ib le to ob ta in a fu l l desc r ip t ion of the

pu r e m e t a l be ha v i ou r , i . e . , t he t h r e e pu r e m e t a l t e r m s , w i t hou t a na l yz i ng a l l oy

da ta .

A n o t h e r w a y o f t r e a t i n g t h e t w o c u r r e n t c o n d u c t i o n h a s b e e n p r e s e n t e d b y

Y a m a s h i t a a nd H a ya ka w a ( 1976) . T he y s t a r t f r om a r e a l i s t i c ba nd s t r uc t u r e m ode l

for Ni and ca lcu la te the res i s t iv i ty by num er ica l ly so lv ing couple d sp in 1' and sp in

$ B o l t z m a nn e q ua t i ons f o r s er i es o f k ve c t o r s . T h e y f ind t ha t t he e l e c t r o n -

m a gnon c on t r i bu t i on t o t he r e s i s t i v i t y i s ve r y s m a l l w he n t he r e i s no i m pur i t y o r




p h o n o n s c a tt er in g b u t b e c o m e s i m p o r t a n t w h e n i m p u r i t y o r p h o n o n s c at te r in g

m a k e s t h e s p i n 1 a n d s p in $ m e a n f r e e p a t h s d i f f e r e n t ; i n t h e l a tt e r c a s e , t h e r e i s

n o m o r e c a n c e l l a t io n b e t w e e n o u t g o i n g a n d i n c o m i n g s c a t te r i n g s . T h i s is

a n o t h e r w a y o f d e s c ri b in g t h e s p in - m i x in g e f fe c t o f t h e m a g n o n s .

2 .2 . Re s i s t i v i t y o f pure me ta l s

2 .2 .1 . Ta bu la r r e su lt s

T h e r e si st iv i ty ( a n d a ls o t h e t h e r m o e l e c t r i c p o w e r a n d t h e r m a l c o n d u c t i v i ty ) o f

p u r e F e , C o a n d N i a r e g iv e n in t a b u l a r fo r m o v e r a w i d e r a n g e o f t e m p e r a t u r e s

b y L a u b i t z e t a l . (1 9 7 3 , 1 97 6 ) a n d a n d F u l k e r s o n e t a l. ( 19 6 6 ).

D a t a o n p o l y c r y s t a ll i n e h e x a g o n a l C o s h o u l d b e t r e a t e d w i t h c a u t i o n , a s t h e

t r a n s p o r t p r o p e r t i e s o f m o n o c r y s t a l s a re h i g hl y a n i s o tr o p i c . A t r o o m t e m p e r a t u r e

( M a t s u m o t o e t a l . 1 9 6 6 )

pc = 10 .3 ixf~cm, Po = 5 .5 ix l Ic m

w h e r e Pc is t h e r e s i s ti v it y m e a s u r e d a l o n g t h e c a x i s w h i l e p p is m e a s u r e d i n th e

p l a n e p e r p e n d i c u l a r t o t h e c d i r e c t i o n . T e x t u r e e f f e c ts i n p o l y c r y s t a l s w i ll c e r ta i n l y

b e i m p o r t a n t .

2 .2 .2 . Res i s t i v i t y a t low t empera ture s

W i th h ig h p u r i t y sa m p l e s a t lo w t e m p e r a t u r e s t h e p r e s e n c e o f t h e i n d u c t io n i n

e a c h f e r r o m a g n e t i c d o m a i n m e a n s t h a t t h e L o r e n t z w O - is n o t n e g li g ib l e e v e n in

z e r o a p p l ie d f ie ld . T h e r e is a n a s so c i a te d i n t e r n a l m a g n e t o r e s i s t a n c e a n d t o g e t

m e a n i n g f u l r e s u l t s f o r t h e i n t r i n s i c l o w f i e ld r e s is t i v i ty th i s e f f e c t m u s t b e e l i-

m i n a t e d a s w e ll a s p o s s i b l e . C a r e f u l e x t r a p o l a t i o n s t o B = 0 u s in g K o h l e r s ' l a w

2Xp/po = f B /p o) h a v e b e e n d o n e f o r N i ( S c h w e r e r a n d S i l co x 1 96 8), C o ( V o l k e n -

s h t e i n e t a l . 1 9 7 3 ) a n d F e ( V o l k e n s h t e i n a n d D y a k i n a 1 9 7 1 ) . F o r F e i n p a r t i c u l a r

t h e i n t e r n a l m a g n e t o r e s i s t a n c e is v e r y i m p o r t a n t , p a rt l y b e c a u s e o f t h e h i g h

v a l u e o f 4 ~ -M a n d p a r t l y b e c a u s e F e b e h a v e s a s a c o m p e n s a t e d m e t a l ( s ee s ec t io n

2 .2 .4 ) s o t h a t th e t r a n s v e r s e m a g n e t o r e s i s t a n c e c a n b e c o m e v e r y s tr o n g , w h e r e a s

t h e l o n g i t u d in a l m a g n e t o r e s i s t a n c e is r e l at iv e l y m u c h w e a k e r . T h i s m e a n s t h a t t h e

o b s e r v e d r e s i s t i v i t y o f a h i g h p u r i t y F e s a m p l e a t l o w t e m p e r a t u r e i s s t r o n g l y

d e p e n d e n t o n t h e d o m a i n c o n f i g u r a ti o n , w h i c h r e g u la t e s h o w m u c h o f t h e s a m p l e

i s s u b m i t t e d t o t r a n s v e r s e m a g n e t i z a t i o n a n d h o w m u c h t o l o n g i t u d i n a l ; t h e

d o m a i n c o n f i g u r a t i o n i s a f u n c t i o n o f a p p l i e d f i e l d , s t r e s s e s a n d m e a s u r i n g c u r r e n t .

T h e r e s is t iv i t y b e h a v i o u r a r i si n g f r o m s u c h e f f e c ts o f i n t e r n a l m a g n e t o r e s i s t a n c e

h a s b e e n s t u d i e d i n d e t a i l in F e w h i s k e r m o n o c r y s t a l s ( T a y l o r e t al . 19 6 8, S h u m a t e

e t a l . 1 9 7 0 , B e r g e r 1 9 7 8 ) ; a n e x a m p l e o f e x p e r i m e n t a l r e s u l t s i s g i v e n i n f i g . 4 . A

c o n t r i b u t i o n t o t h e r e s i s ti v i ty f r o m t h e i n t e r n a l H a l l e f f e c t h a s a l s o b e e n f o u n d i n

C o m o n o c r y s t a l s ; t h is c o n t r i b u t i o n is a s s o c i a t e d w i t h t h e z i g -z a g p a t h o f t h e

c o n d u c t i o n e l e c t ro n s w h i c h i s i n d u c e d b y t h e H a l l e f f e ct in a p o l y d o m a i n s a m p l e

( R a m a n a n a n d B e r g e r 1 9 7 8 , B e r g e r 1 9 7 8 ) .

T h e l o w t e m p e r a t u r e r e si s ti v it y o f N i , C o a n d F e h a s b e e n f o u n d t o v a r y a s



T R A N S P O R T P R O P E R T I E S O F F E R R O M A G N E T S 763

1800

1600-

~ 1400

1200-

g

N 1000-

800

~3

~- 0

-0.2 -

-0 4

0_

~-- -0.6

-0.8

~ { L

oo

100 200 300 400 500 600 700 800 Oe

Magnetic FieLd

Fig. 4. Mag netizat ion of iron single crystals above) and magn etores istance below) of 100) and 111)

iron wh isker s at 4.2 K as a function of applied field after Tay lor et al. 1968).

p o + A T 2 by most w orkers; above 10 K an additional term in T 4 is generally

needed (White and Woods 1959, Greig and Harrison 1965, White and Tainsh

1967, Schwerer and Silcox 1968, Beitcham et al. 1970). In early work no mag-

netoresistance corrections were made, with the result that rather varied values of

A were obtained. In a number of samples a term linear in T was also needed, but

careful experiments on the effect of magnetic fields show that this linear term is

not intrinsic but is a result of the internal magnetoresistance (Volkenstein et al.

1971, 1973).

The best values of A from data to which the magnetoresistance correction was

applied to obtain values at B = 0 are:

A = 9.5 x 10 -12 ~ c m K -2 for Ni (Schwerer and Silcox 1968)

A = 16 × 10-12 l ~cmK 2 fo r C o (Volk ens te in et al. 1973)

A = 15 × 10-12 ~ c m K -2 for Fe (Volken ste in et al. 1971, 1973).

These values however are not quite consistent with measurements on other

samples, even if internal magnetoresistance effects are taken into account. It is

likely that

p T )

depends in some way on the nature of the residual impurities. As

the sca ttering by residual impuritie s is still p red omin ant up to 10 K in the pu rest

samples,

p T )

is actually expected to show deviations from Matthiessens' rule

similar to those observed in alloys and explained in the two-current model



764 I .A. CAM PBELL AND A. FER T

( s e c t i o n 2 . 3 ) . M o r e p r e c i s e l y , t h e t w o - c u r r e n t m o d e l r e l a t e s

p T )

t o th e p a r a m e t e r

a c h a r a c t e r i s t i c o f t h e i m p u r i t y s c a t t e r in g ( e q . (2 5 )) . A c c o r d i n g t o w h e t h e r a is

l a r g e o r c l o s e t o u n i t y

p T ) - p o

is l a r g e a n d v a r y i n g a s p , + ( T ) o r s m a l l a n d

v a r y i n g a s

pi T ) .

T h e v a l u e s o f

p T ) - p o

f o r p u r e m e t a ls , a l t h o u g h s c a t te r e d , a r e

r e l a t i v e l y s m a l l , w h i c h s u g g e s t th a t a is g e n e r a l l y c lo s e t o u n i t y . T h e v a r i a t i o n i n

T 2 c a n b e t h e n a s c r i b e d t o

p i T ) .

T h e v a r i a t i o n in T 2 h a s b e e n a t t r i b u t e d e i t h e r t o e l e c t r o n - m a g n o n s c a t te r in g o r

t o s - d e l e c t r o n - e l e c t r o n s c a t t e r in g ( B a b e r 1 9 37 ) o f t h e s a m e t y p e a s l e a d s t o a T 2

t e r m in t h e re s i s ti v i ty o f n o n - m a g n e t i c tr a n s i t i o n m e t a l s a t l o w t e m p e r a t u r e s , a n d

w h i c h i s m u c h t h e s a m e m a g n i t u d e a s th e T 2 t e r m in t h e f e r r o m a g n e t s . I f th e

t h e r m a l c o n d u c t i v i ty o f t h e f e r r o m a g n e t i c m e t a l s is al so m e a s u r e d a t lo w t e m -

p e r a t u r e s t h e L o r e n t z r a t i o c o r r e s p o n d i n g t o t h e n o n - i m p u r i t y s c a t t e r i n g i s a b o u t

1 × 1 0 8 W ~ ) K -2 b o t h i n N i ( W h i t e a n d T a i n s h 1 9 67 ) a n d i n F e ( B e i t c h a m e t a l .

1 9 7 0 ) . T h i s i s c l o s e t o t h e v a l u e e s t i m a t e d t h e o r e t i c a l l y ( H e r r i n g 1 9 6 7 ) f o r s - d

e l e c t r o n - e l e c t r o n s c at te r in g . H o w e v e r t h e w a y in w h i c h t h e e x p e r i m e n t a l d a t a a r e

a n a l y z e d h a s b e e n c r i ti c i z ed ( F a r r e l a n d G r e i g 1 9 69 ). S e c o n d l y , w e c a n c o n s i d e r

d a t a o n

p i t T ) , p i + T )

a n d p ~ + ( T ) o b t a i n e d f r o m a n a n a l y si s o f d il u t e N i a l l o y s

( s e e s e c t i o n 2 . 3 ). T h e s p in - fl ip m a g n o n - e l e c t r o n r e s is t iv i t y p~ + ( T ) is r o u g h l y

5 × 1 0 9 l ~ c m a t 1 0 K i n N i ( s e e fi g. 9) ; b e c a u s e o f t h e s m a l l a n g l e s c a t t e r i n g f a c t o r

t h e e l e c t r o n - m a g n o n c o n t r i b u t i o n s t o p ~, ( T ) , p i+ ( T ) s h o u l d b e l o w e r b y a f a c t o r

o f t h e o r d e r o f

T/Tc,

g iv in g pi~ ( m a g n o n s ) - 1 0 - 1 ° ~ c m ; a s t h e o b s e r v e d pi~

v a l u e s i n N i a t 1 0 K a r e m u c h h i g h e r ( t h e s e a r e h i g h e r t h a n

10 9

~~cm ) , we can

i n f e r t h a t t h e e l e c t r o n - m a g n o n c o n t r i b u t i o n s t o t h e p~r a r e n o t d o m i n a n t . T h i s

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

s t r u c t u r e m o d e l o f N i ( Y a m a s h i t a e t a l. 1 9 75 ). I t t h u s t u r n s o u t t h a t e l e c t r o n -

e l e c t r o n c o l li si o ns p l a y t h e m a j o r r o l e i n t h e l o w t e m p e r a t u r e T 2 t e r m o f t h e p u r e

m e t a l s .

2.2.3. Residual resis t iv i ty

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

r e s is t iv i t y w h i c h i s n o l o n g e r p r o p o r t i o n a l t o t h e c o n c e n t r a t i o n o f i m p u r i t i e s . T h i s

e f f e c t is p a r t i c u l a r l y i m p o r t a n t f o r F e w h i c h h a s a h i g h v a l u e o f 4 7 r M a n d a h i g h

t r a n s v e r s e m a g n e t o r e s i s t a n c e (fig . 5 ). F o r d e m a g n e t i z e d F e p o l y c r y s t a l s it w a s

p o i n t e d o u t t h a t t h e a p p a r e n t r e s i d u a l r e s i s ti v i ty r a t i o p ( 3 0 0 K ) / p ( 4 . 2 K ) w o u l d

n e v e r i n c re a s e b e y o n d a b o u t 3 0 0 h o w e v e r p u r e t h e s a m p l e ( B e r g e r a n d d e

V r o o m e n 1 9 6 5 ) . I t i s n o w s t a n d a r d p r a c t i c e t o m e a s u r e t h e l o w t e m p e r a t u r e

r e s is t iv i t y o f F e s a m p l e s i n a s a t u r a t i n g l o n g i t u d i n a l m a g n e t i c f i e ld s o a s to

e l i m i n a t e t r a n s v e r s e m a g n e t o r e s i s t a n c e . T h i s c a n r e d u c e t h e a p p a r e n t r e s i s t i v i t y

b y a f a c t o r o f 5 o r m o r e . I n p r i n c i p l e a c o r r e c t i o n s h o u l d s t i l l b e m a d e f o r t h e

l o n g i tu d i n a l m a g n e t o r e s i s ta n c e . I n N i s a m p l e s t h e e n h a n c e m e n t o f t h e r e s id u a l

r e s is t iv i t y b y t h e i n t e r n a l m a g n e t o r e s i s t a n c e is l es s i m p o r t a n t t h a n i n F e b u t s ti ll

s i g n i f i c a n t ( F u j i i 1 9 7 0 , S c h w e r e r a n d S i l c o x 1 9 7 0 ) .

T h e c o n t r i b u t i o n o f t h e d o m a i n w a ll s t o t h e r e s i d u a l re s i s ti v i ty o f f e r r o m a g n e t i c

m e t a l s h a s b e e n s u b j e c t t o m a n y d i s cu s s io n s . I t n o w a p p e a r s t h a t d o m a i n w a l l s a r e

t o o t h i c k t o s c a t t e r e l e c t r o n s a p p r e c i a b l y . H o w e v e r , a s i t h a s b e e n p o i n t e d o u t i n




1000 -

x

500

B nO

0

o xrl

x x O O

O V ^Jm O 0 A

c~ 200 n

.4.. O n

~ 0

100 --

c l

0... x[3

X

50 x

°

0 *o I n c r e a s i n g p u r i t y

20

x I , i i , ~ ,, I I , i I L, ,, I ,

20 50 100 200 500 1000 2000

((3295 / (34 .2 )mox .

Fig. 5. Residual resistance rat io in zero field and in longitudinal applied field for i ron samples of

increasing puri ty after Berger 1978).

t he preceding sec t ion , the res i s t iv i ty depends ind i rec t ly on the domain wa l l s a s i t

de pe nds on t he dom a i n c on f i gu r a t i on i n t he s a m p l e ( B e r ge r 1978 ) .

2 2 4 High field behaviour

I t is w e l l know n t ha t t he m a gne t o r e s i s t a nc e a nd H a l l e f fe c t o f pu r e m e t a l s un de r

h igh f i elds such tha t wc~ >> 1 g ive in fo rm at io n on the Ferm i sur face .

I n N i unde r h i gh f i e l d s a pp l i e d a l ong a non - s ym m e t r y d i r e c t i on o f a m onoc r ys -

t al , the t r a ns ve r s e m a gne t o r e s i s t a nc e s a t u ra t e s , a nd R 0 c o r r e s p ond s t o a n e f f e c ti ve

c a r r i e r de ns i t y o f 1 e l e c t r on pe r a t om , e ve n t hough N i ha s a n e ve n num be r o f

e l e c t r o n s a n d s o w o u l d n o r m a l l y b e e x p e c t e d t o b e h a v e a s a c o m p e n s a t e d m e t a l .

R e e d a nd F a w c e t t ( 1964a ) s how e d t ha t a f e r r om a gne t i c m e t a l d i d no t ha ve t o obe y

t he s a m e r u l e s a s non - m a gne t i c m e t a l s be c a us e o f t he i ne qu i va l e nc e o f s p in 1' a nd

s p in $ . T h e y de d uc e d f r om t he i r r e su l t s t ha t t he m i no r i t y d ba n d i n N i w a s

e l e c t r on - l i ke i n c ha r a c t e r . T he be ha v i ou r o f t he m a gne t o r e s i s t a nc e f o r c e r t a i n

f i el d d i re c t i ons i nd i c a t e d t he p r e s e n c e o f ope n o r b i t s f o r c e r t a in f i e ld o r i e n t a t ions .

T h e r e s u l t s c o u l d b e c o m p a r e d w i t h d e H a a s - V a n A l p h e n d a t a ( H o d g e s e t a l .

1967 , Tsui 1967 , Ruva lds and Fa l i cov 1968) . No obvious t rans i t ion cor responding

t o a m a j o r d i f f e r e nc e i n m ob i l i t y fo r d - l i ke a nd s - l ike pa r t s o f t he F e r m i s u r f a c e

w a s obs e r ve d .

I n C o t he t r a ns ve r s e m a gne t o r e s i s t a nc e i s a ga i n s a t u r a t e d ( C o l e m a n e t a l . 1973 )

w i t h ope n o r b i t be ha v i ou r f o r c e r t a i n s pe c i a l d i r e c t i ons .

I n F e u p t o f ie ld s o f a bou t 10 0kG , t he m a gne t o r e s i s t a nc e t e nds t o a B 2

de pe nde nc e i nd i c a t i ng t ha t t he m e t a l i s c om pe ns a t e d ( R e e d a nd F a w c e t t 1964b) .

Th er e app ears to b e a con s ide ra b le sprea d o f o)c~- va lues . In the same f i e ld range

a t l ow t e m p e r a t u r e s t he o r d i na r y H a l l c oe f f ic i e n t R 0 i s s t r ong , ne ga t i ve a nd



7 66 I .A . C A M P B E L L A N D A . F E R T

weakly field dependent (Klaffky and Coleman 1974). This also is consistent with a

metal having compensated character, for which R0 bears no relationship to any

effective number of electrons per atom.

At stil l higher values of ~o~- the mag netoresista nce increases much more slowly

than B 2 (Colem an 1976); magnetic breakd own and intersheet scattering have been

invoked.

2 _3 A llo ys: residual

r esistivi ty nd temper tur e dependence

of resistivity

The residual resistivity per atomic percent impurity has been measured for a wide

range of impurities in Ni, Co and Fe (p0 in tables 1, 2, 3) and the deviations from

Matt hiessens rule have be en stu died both for ternary alloys (fig. 6) and for binary

alloys as a function of temperature. Using the two current model equations of

section 2.1.2 the experimental data have been used to determ ine the spin t and

spin ~, residua l resistiv ities Pot and p0~ for each impu rity in each ho st (ta bles 1, 2,

3 .

T A B L E 1

V a l u e s o f

a =

P o l / P o t p o

Pot, Po,,

for d i lu te impuri t i e s in n icke l .

Po po t Po

Impuri ty in n icke l c~ = Po ~/Po

t ( i xf~cm) ( ix~c m) (~ f~cm )

C o 13 ( ), 300,), 20 (c),

13(d) 20(f), 20(g 0. 16 ± 0.0 3 0. 18 --+ 0.0 3 3. 5 --+ 1

Fe 11 ( ), 200,), 7.3 (d) 0. 36 -+ 0. 04 0.4 -+0 .04 6-+ 1.5

M n 6.3 ( /, 150,), 5.4 (d) 0.7 -+ 0.1 0.7 5 -+ 0.2 7.5 + 2.5

C r 0.2 1 ( ), 0.45 (b), 0. 4 (c),

0.21(a), 0.2#), 0.4(g 5 -+ 0.1 22 -+ 6 6.5 -+ 0. 5

V 0.45 ( ), 0.55 °,), 2.3 (d) 4. 4-+ 0.2 13-+ 1 6.7 -+ 0.5

Ti 0.9 (a), 40,), 2.7 (d) 3.3 -+ 0.6 5.6 -+ 2 10.5 -+ 4

Pd 1 d) 0.2 -+ 0.0 5 0.3 Id) 0.3 (a)

R h 0.3 ( ), 0.1 7 ( ), 0.290) 1.8 _+0.1 10 _+ 2 2.1 _+ 0.2

R u 0.0 75 ( /, 0.1 5 (e) 4.8 _+ 0.2 56 -+ 15 5.8 -+ 0.5

M o 0.28 e), 0.37 °) 6.4 -+ 0.6 25 + 4 8 + 1

N b 0.4 4 ( /, 0.470) 5 _+ 0.2 16 -+ 1 7 _+ 0.2

Z r 7.5 (e) 2.8 ± 0.5 4 (e) 30 (e)

Pt 0.2 4 (a), 0.1 7 (~) 0.85 ± 0.2 5.3 ± 1.6 1 _+ 0.2

Ir 0.24 ( ), 0.13 (~) 3.8 ± 0.2 28 ± 7 4.8 ± 0.2

O s 0.13 ( ~, 0.13 o) 5.5 _+ 0.5 50 ± 2 6.4 ± 0.5

R e 0.3 ( ) , 0.26 e) 5.8 ± 0.5 26 ± 3 7.5 ± 0.5



T A B L E 1 ( c o n t in u e d )

Po po t Po ,~

I m p u r i t y i n n i c k e l a = po ~ Po t (txf~cm) (ixf~cm) (pf ~cm )

W 0.4 (e), 0.5 o) 6 ± 0 .5 16.5 -+ 1 7 -+ 0.5

T a 0.53 (e), 0.46 o) 5.2 ± 0.5 16 ± 1 7.5 ± 0.5

H f 8.6 (~), 8.1 o) 3.6 ± 0.5 3.5 ± 0.5 30 - 1

Cu 2.9 (a), 3.7 (d) 0.9 ± 0.1 1.1 ± 0.2 3.7 ± 0.2

A u 5.9 ( ) 0.36 ( ) 0.44 (a) 2.6 (a)

A I 1.7 a) 2.13 (a) 3.4 (a) 5.8 (a)

Si

1.3 ) 2.83 °) 5 (a) 6.4 (a)

Zn 2. 2 ~) 1 ± 0.1 1.3 a) 2. 9 a)

G a 1.7 g) 1.91 (g) 3cg) 5. 2 (g)

G e 1 g) 2. 84 (g/ 5.7 (g/ 5. 7 (g)

In 1.50 ) 3.60 ) 6 °`) 90 )̀

Sn 1.6 a), 1.350 ) 3.2 ± 0.4 5.2 -+ 0.8 7.7 ± 0.5

Sb

0.8 ~)

1.6c~)

3.60') 2.90')

* F o r a w e g iv e t h e v a lu e s f o u n d b y :

a)

Dor le i jn and M ied em a (1975a) , Dor le i jn (1976) ; (0 Cadevi l le e t a l. (1968) ;

0`) Fer t an d Ca mp be ll (1976); ~) Hu ge l (1973);

(c) Le on ar d e t a l . (1969); 0 ) Ro ss e t a l . (1978);

ca) Farre ll an d Gre ig (1969); (i) D ur an d (1973).

(e) D ur an d and Ga ut i e r (1970) ;

T h e v a lu e s o f a g iv e n i n (a ), ( b) , (f ), (g ) h a v e b e e n m o s t ly d e r iv e d f r o m m e a su r e m e n t s o f t h e

res idua l r e s is t iv i ty of te rna ry a l loys , whic h i s the m ost d i rec t me tho d. T he v a lues of a g iv en in (d), ( e) ,

( h) , (i ) h a v e b e e n o b t a in e d o n b in a r y a l l o y s f r o m th e d e v i a t i o n s f r o m Ma t th i e s se n s r u l e a t l o w

te m p e r a tu r e ( h ), 7 7 K ( i) , 3 0 0 K w i th t h e a s su m p t io n o f c o m p le t e sp in m ix in g ( d ) o r 30 0 K w i th t h e

a s su m p t io n o f n o sp in m ix in g ( e ). T h e v a lu e s t h a t w e g iv e f o r p0 , P o t , p0 ~ h a v e b e e n e s t im a t e d f r o m

th e sp r e a d o f t h e v a lu e s f o u n d in t h e l i t e r a tu r e .

2

o_£ 1

0

// Ni (Co1_xRhx)

05

1

lo

N.~ (AUl _xCO x)

0 5[

1

0 015 1.0

X

F ig . 6 . R e s id u a l r e s i s t i v i t i e s o f N i ( C O l - x R h x ) a n d N i ( A u l x C o x ) a l l o y s . T h e l a r g e d e v i a t i o n s f r o m

Ma t th i e s se n s r u l e ( b r o k e n l i n e) f o r t h e N i ( C O l - x R h x ) a l l o ys a r e a c c o u n te d f o r b y v e r y d i f f e r e n t v a lu e s

of C~co and aRh; the so l id c urve i s ca lcu la ted f rom eq . (24) wi th aco = 13 and aRh = 0 .3 . The ve ry smal l

d e v i a t i o n s f r o m

M R

i n t h e N i ( A u l - x C o x ) a r e a s so c i a t e d w i th v a lu e s o f a A u a n d aC o b o th m u c h l a r g e r

tha n 1 (a f te r Dor le i j n 1976).

767



768

I.A. CAMPBELL AND A. FERT

TABLE 2

V a l u e s o f a = Po ~/po ~, po, Po t, po ~ f o r d i l u t e i m p u r i t i e s i n c o b a l t * .

Po P o ~ p o

I m p u r i t y i n c o b a l t c~ = Po /Po ~ 0xf~cm) (Ixf~cm) (~ftcm)

Fe ( ) 12 0.5 0.54 6,7

Mn c°/ 0.8 5.5 12 10

Cr e°} 0,3 1.8 7.3 2.4

V (b) 1 3.8 7.7 7.7

Ti (u~ 1.4 4.5 7.6 11

R h c°~ 1 1.4 2.8 2.8

Ru ( / 0.22 4.0 22.4 4,86

Mo c°~ 0.7 6.0 14.4 10

Nb ~ 1 6.5 13 13

Zr c°~ 3.3 4.0 5.2 17

Ir a~ 0.33 2.9 11.7 3.82

Os(a~ 0.29 5,3 23.5 6.84

R e c° )

0.43 5.3 18 7.7

W (b~ 0.84 5.7 10.5 12.5

T a c°~

1.23 5.5 10 12.3

H f (b) 2.5 4.0 5.5 14

Sn (c~ 1.2 2.9 5.3 6.4

Sb (c) 0.9 2 4.2 3.8

• ca)L o e g e l a n d G a u t i er 1971; ~b~Durand 1973; to)R o s s e t al. 1978. The d a t a

h a v e b e e n o b t a i n e d f r o m d e v i a t i o n s f r o m M a t t h i e s s e n s r u l e i n t h e r e s id u a l

r e s i s t i v i t y o f t e r n a r y a l l o y ( a ) o r i n t h e r e s i s t i v i t y o f b i n a r y a l l o y s a t l o w

t e m p e r a t u r e ( c ) o r a t 77 K (b). We h a v e p r e f e r r e d t h e r e s u lt s g iv e n b y D u r a n d

(1973) t o s li g h t ly d i ff e r en t o n e s g i v e n p r e v i o u s l y b y D u r a n d a n d G a u t i e r

(i970).

2 3 1 Nickel ost

h e g e n e r a l p i c t u r e o f c~ ( = p 0 ~ / p 0 t ) v a l u e s f o r im p u r i t ie s i n N i e s t im a t e d b y

d i f fe r e n t g r o u p s is c o n s i s te n t , a l t h o u g h n u m e r i c a l v a l u e s a r e n o t i n p e r f ec t

a g r e e m e n t ( t a b l e 1 , f ig . 7) . It is f o u n d t h a t C o , F e , M n , A u a n d C u h a v e a >> 1

w h i l e C r , V a n d a n u m b e r o f o t h e r t r a n s i ti o n im p u r i t ie s h a v e a < 1 . A s h a s b e e n

p o i n t e d o u t ( D u r a n d a n d G a u t i e r 1 9 7 0, F er t a n d C a m p b e l l 1 9 71 , 1 9 76 , H a g a k a w a

a n d Y a m a s h i t a 1 9 7 5 ) t h e r e is a v e r y c le a r c o n n e c t i o n b e t w e e n t h e e l ec t r ic a l a n d

t h e m a g n e t i c p r o p e r t i e s o f t h e i m p u r i t ie s . T h o s e i m p u r i t ie s w i t h h i g h v a l u e s o f o~

a r e t h o s e w h i c h , o n t h e F r i e d e l a n a l y s is o f t h e m a g n e t i c p r o p e r t i e s ( F r i ed e l 1 9 6 7) ,

d o n o t h a v e d I' v ir t u a l b o u n d s t a t e s a t o r n e a r t h e s p i n 1' F e r m i e n e r g y . T h e s e

i m p u r i t ie s h a v e l o w P 0 t v a l u e s b e c a u s e t h e d 1" p h a s e s h i f t a t t h e F e r m i e n e r g y i s

s m a l l ; i n c o n t r a s t , w h e n t h e i m p u r i t y i s s u c h t h a t a d ] ' v i r t u a l b o u n d s t a t e i s c l o s e

t o t h e F e r m i e n e r g y , P o t i s l a r g e s o a i s s m a l l . T h e d i f f e r e n c e i n Po t v a l u e s

b e t w e e n t h e s e t w o t y p e s o f i m p u r i t i e s c a n b e q u i t e s t r i k in g : P o t -~ 0 . 1 6 I xf~ cm /%

f o r C o i m p u r it ie s w h i le P o t = 5 6 1 x ~ c m / % f o r R u i m p u r i ti es D e t a i l e d c o m -

p a r i s o n s b e t w e e n c a l c u l a t e d a n d e x p e r i m e n t a l v a l u e s o f s p i n 1' a n d s p in

r e s i s t i v i t i e s h a v e b e e n m a d e .

T h e t e m p e r a t u r e d e p e n d e n c e o f t h e re s is t iv i ty o f b i n a ry a l lo y s o f N i ca n b e



T R A N S P O R T P R O P E R T I E S O F F E R R O M A G N E T S

T A B L E 3

V a l u e s o f a =

PoUPot , Po ,

Pot , P01 for d i lu t e impur i t i e s in Fe* .

769

Po Po i' Po $

I m p u r i t y i n F e oL = po ~ Ipo

t

(Ixl~cm) ( ixf~cm) (Ixf~cm)

N i 3 (a), 70 ) 2 _+ 0. 2 2. 4 _+ 0. 2 12 _+ 5

C o 1 ), 3.70') 0.9 _+ 0.1 1.6_+ 0.4 3.3_+ 1.3

M n 0.0 9 (a), 0.170') 1.5 _+ 0.2 13 _+ 5 1.7 _+ 0.2

Cr 0.17 ( ) , 0 .37 (8) 2.2_+0.3 12.5_+6 2.8_+0 .2

V 0.12 C ), 0.13 °') 1.1_+ 0.3 10.5_+ 3 1.3_+ 0.3

Ti 0.2 5 (a), 0.6 6 (8) 2.7 5 _+ 0.2 5 10.5 _+ 4 4 _+ 0.4

R h 5.8 (8) 0.95 ~ 1.1 °')

6.4 0')

R u 0.380') 2 _+ 0.1 7. 3 °') 2.80')

M o 0.210') 1.75 ~ 0.2 110') 2.30')

Pt 80') 1.3 (8) 1.5 °') 120')

I r 90') 2 (8) 2. 2 °') 200')

O s 0.33 °') 3.5 _+ 0.5 130') 4.3 °')

R e 0.31 c°) 2.7 _+ 0.5 8.7 °') 2.7 (8)

W 0.240') 1.6 _+ 0.1 7.50') 1.8 °')

Be 6.2 °') 40') 4.7 °') 29 °')

A1 8.6 °') 5.3 _+ 0.2 5.6 °') 48 °')

Si 5.60') 6 _+ 0. 6 6.40') 36 °')

G a 8.1 (8) 4.8 °') 5.40') 440')

G e 6.2 °') 6.8 _+ 0.2 7.9 (8) 49 °')

Sn ~ 1 c) 8. 7 _+

Sb ~ 1 (~) 9.8 _+ 0. 4

* F o r a w e g i v e t h e v a l u e s d e r i v e d b y : (~) F e r t a n d C a m p b e l l ( 1 9 7 6 ) ; 0 ) D o r l e i j n

an d M ied em a (1977), Dor l e i jn (1976); (C)Ross e t al . 1979 , f ro m the re s idu a l

r e s i s t i v i t y o f t e r n a r y a l l o y s ( b ) o r f r o m p T ) o f b i n a r y a l l o y s ( a ) a n d ( c ) . W h e n

t h e r e a r e d a t a f r o m s e v e ra l a u t h o r s w e h a v e e s t i m a t e d m e a n v a l u e s o f p0 , p 0 t ,

Po~.

I

E 20.

15

0_.-

~_~ 10

c~

J

Z

I I I

Ti V Cr iqn Fe C o Ni

F i g. 7 . S u b - b a n d r e s i d u a l r e s i s t iv i t i es p 0 t a n d p 0 ~ o f 3 d i m p u r i t i e s i n n i c k e l . ( R e f e r e n c e s i n f o o t n o t e t o

t a b l e 1 . )



770 I.A. CAMPBE LL AND A. FERT

a n a l y z ed b y u s in g t h e t w o - c u r r e n t m o d e l e q u a t i o n s t o e s ti m a t e t h e p u r e m e t a l

pa r a m e t e r s P t ~ (T ) , Pit (T ) a nd p i , ( T ) . F i gu re 8 s how s t he a g re e m e n t be t w e e n

e xpe r i me n t a l r e s u l ts be l ow 50 K fo r s e ri e s o f N i a ll oys a nd c u rve s ob t a i ne d f rom

e q . (25) by us ing a va l ue s de r i ve d f rom i nde pe n de n t me a s u re me n t s on t e rn a ry

alloys, /x = 3.6,

pi(T)

= 9.5 x 10-12T2+ 1.7x

10-1 4T 4

( in f~cm i f T is expressed in

K) and P t +(T) of fig. 9 (dashed l ine ) . At t e m pe ra t ur es up to abo ut 50 K the

a na l ys i s c a n be done una mbi guous l y bu t a t h i ghe r t e mpe ra t u r e s d i f f e r e n t s e t s o f

so lu t ions f i tt ing the ex per im enta l da ta a re poss ib le . At 300 K a reas ona ble

est im ate is Pt ~(300) = 11 tx~c m, pit (300) = 6.7 tM2cm, pi~(300) = 27 FxlIcm (Fe rt

and Campbe l l 1976) .

T h e c o n t r i bu t i on t o P t ~ (T3 f rom e l e c t ro n -m a gn on c o ll is i ons ha s be e n c al -

cul a ted by Fe rt ~(1969) and Mil ls e t a l . (1971) in a mo de l of spin-spl i t sphe rical

F e rmi s u r f a c e s . T he c a l c u l a t i on g i ve s t he c o r r e c t o rde r o f ma gn i t ude . T he

va r i a t i on ob t a i ne d fo r p~ (T )/ T 2 as a funct ion of T is shown in f ig. 9 (sol id l ine)

t oge t h e r w i t h t he va r i a t i on n e e d e d t o f it t he e x pe r i m e n t a l r e s u l t s ( da s he d l i ne ).

T he c a l c u l a t e d c u rve d rops a t l ow t e mpe ra t u r e , w h i c h r e s u l t s f r om a f r e e z i ng ou t

o f e l e c t r on -m a gn on , s c a t t e r ing in t he p r e s e nc e o f a ga p be t w e e n s p in I a nd s p in

F e rmi s u r f a ce s ; t he e xp e r i me n t a l c u rve s how s a s imi l ar d rop be l ow a bou t 30 K

a nd t he n a n up t u rn be l ow 5 K ; t h i s up t u r n s e e ms t o be a s s oc i a te d t o a va r i a t ion i n

T 3/2 a t ve ry l ow t e mpe ra t u r e a nd ha s be e n a s c r i be d t o e l e c t ron -ma gnon s c a t t e r i ng

10

4

~T (10-115~cm

°K-2

x × x

{CxC °

x

/~ /~: atc . _. o

/ * / N i M n O A * / , d

,¢ / / o /O

? //.x./

x x r o co le a

x - ~

,, x ~ /7 7 o °

/_~a_NiCrt6°lo~

' -u tx

_ M~ cole.

t~

10 20 30 40 50

Fig. 8. pr/T 2= Co T)-p O))/T 2 against T for several nickel based alloys. The solid curves are

calculated f rom eq. 25) in the way described in the text after Fert and Campbel l 1976).




in reg ions wh ere the sp in 1' and sp in $ Ferm i sur faces touc h or a re ve ry nea r

(Fer t and Campbe l l 1976) .

The res i s t iv i t i e s

pi~(T)

a r e e x p e c t e d t o i n c l u d e c o n t r i b u t i o n s f r o m e l e c t r o n -

e l e c t r o n , e l e c t r o n - p h o n o n a n d e l e c t r o n - m a g n o n c o l l i s i o n s . B e c a u s e o f t h e s m a l l

a ng l e s c a t t e r i ng f a c t o r t he e l e c t r on - m a gnon c on t r i bu t i ons t o p i ~ , p i + s hou l d e a c h

b e e q u a l t o r o u g h l y (T/Tc)p~ + a nd t he r e f o r e r e l a t i ve l y s m a l l a t l ow t e m pe r a t u r e s .

I f t h e n t h e e l e c t r o n - e l e c t r o n o r e l e c t r o n - p h o n o n c o n t r ib u t i o n s a r e d o m i n a n t , t h e

poss ib i li ty of s ca t t e r ing of the sp in $ e l ec t ro ns to the d $ ban d m ake s

pi (T)

l a r ge r t ha n pi~(T), i n a g r e e m e n t w i t h /~ > 1. A t ve r y l ow t e m pe r a t u r e s t he

va r i a t i on o f

pi(T)

in T 2 c a n b e a t t r i bu t e d t o e l e c t r o n - e l e c t r on s c a t te r i ng , a s i t ha s

be e n c onc l ude d i n s e c t i on 2 . 2 . 2 . A bove 10K t he e l e c t r on - phonon c o l l i s i ons

b e c o m e p ro g r e s si v e ly m o r e i m p o r t a n t . W h e n a p p r o a c h i n g r o o m t e m p e r a t u r e t h e

e l e c t r on - m a gn on c o l li s ions s hou l d be g i n t o m a ke a s ubs t a n ti a l c on t r i bu t i on t o

pi~(T). T he o r e t i c a l e s t im a t e s o f t he e l e c t r o n - p ho no n c on t r i bu t i on s t o P it a nd pi+

a t 300 K a r e 4 . 25 ~ c m a nd 19.2 ~l ~cm r e s pe c t i ve l y ( Y a m a s h i t a a nd H a ya ka w a

1976) ; w e c a n r e a s ona b l y in f e r t ha t a dd i t i ona l c on t r i bu t i ons o f a f e w ~ c m f r om

e l e c t r on - m a gnon s c a t t e r i ng a c c oun t f o r t he e xpe r i m e n t a l p i ~ ( 300 ) . Wi t hou t m a g-

non con t r ibu t ions to p i~ (300) and wi th out p~ ~ t e rm the res i s tiv i ty of pure n icke l a t

3 0 0 K w o u l d b e p r e d i c t e d t o a m o u n t t o r o u g h l y 4 . 2 5 x 1 9 . 2/ (4 . 2 5+ 1 9 . 2 ) -

3 .5 ~ c m , i n s te a d o f 7 p ~ c m e x p e r i m e n t a ll y . W e c o n c l u d e t h a t:

( i ) a t l ow t e m pe r a t u r e t he m a i n c on t r i bu t i ons t o p~(T) a r i s e f r o m e l e c t r o n -

e l e c t r o n a n d e l e c t r o n - p h o n o n s c a t t e r i n g s ; e l e c t r o n - m a g n o n c o l l i s i o n s c o m e i n t o

p l a y t h r ough p~ ~ ( T ) a nd a r e i m por t a n t i n a ll oys w i t h ~ ve r y d i f f e r e n t f r om un i t y ;

(ii) a t n e a r r o o m t e m p e r a t u r e t h e e l e c t r o n - m a g n o n c o ll is io n s c o n t ri b u t e t o

bo t h pi~ a nd p~ +; t he y w il l be c o m e i nc r e a si ng l y i m p or t a n t a s t e m pe r a t u r e

inc reases .

T h e a n al ys is o f th e e x p e r i m e n t a l d a t a o n N i a ll oy s b y Y a m a s h i t a a n d H a y a k a w a

( 1976) , a l t hough ba s e d on a d i ff e r e n t t r e a t m e n t o f t he t w o c u r r e n t c on duc t i on ,

a r r ives a t s imi la r conc lus ions .

2.3.2. Coba lt host

T h e o~ va l ue s o f a l a rge nu m b e r o f im pur i t i e s ha ve be e n o b t a i ne d i n C o m e t a l

( D ur a nd a nd G a u t i e r 1970 , L oe ge l a nd G a u t i e r 1971 , D ur a nd 1973 , R os s e t a l .

1978) , t ab le 2 . They a re aga in cons i s t en t wi th the magne t i c s t ruc tures of the

i m p u r it ie s . T h e p a r a m e t e r s Pit (T ), pi+ (T ) a n d p , + T ) o f C o h a v e b e e n e v a l u a t e d

by L oe g e l a nd G a u t i e r ( 1971) ; t he b e ha v i ou r o f p~ ~(T) i s s imi la r to tha t o f Ni .

2.3.3. Iron host

E xt e ns i ve w or k ha s be e n done on F e ba s e d a l l oys ( C a m pbe l l e t a l . 1967 , F e r t a nd

Campbe l l 1976 , Dor le i jn 1976 , Dor le i jn and Miedema 1977, Ross e t a l . 1979) ,

t a b l e 3 . T h e r e s u l ti ng ~ va l ue s f r om d i f f e r e n t a u tho r s , bo t h f r om t e r na r y a l l oy

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

othe r . T he rang e of c~ va lues i s ve ry gre a t , po ~/po ~ va rie s f ro m 0.13 f o r F__eeV o 9 fo r

F e I r ( t ab l e 3 ). T he r e is a good c o r r e l a t i on b e t w e e n t he r e s is t iv i t y p0 i n e a c h ba n d

a nd t he c ha r ge s c r e e n i ng i n t ha t ba nd f o r e a c h i m pur i t y ( D or l e i j n 1976 ) .



7 7 2 I .A . C A M P B E L L A N D A . F E R T

,_NC

T

C

v

k -

12

- 4 P

J

10 20 30 40 50

Temp6rature K)

F i g . 9 . E x p e r i m e n t a l d a s h e d l i n e ) a n d c a l c u l a t e d s o l id l i n e ) c u r v e s f o r p , ~ / T 2 i n n i c k e l . T h e

e x p e r i m e n t a l c u r v e is a f t e r F e r t a n d C a m p b e l l 1 97 6) ; t h e c a l c u l a t e d c u r v e i s o b t a i n e d f r o m t h e m o d e l

c a l c u l a t i o n o f F e r t 1 96 9 ) w i t h 01 = 3 8 K .

T h e b e h a v i o u r o f p , + ( T ) i n F e i s s i m i l a r t o t h a t i n N i a n d t h e v a l u e o f

pi+/pi,

s e e m s t o b e n e a r 1 ( F e r t a n d C a m p b e l l 1 97 6). M o r e c o m p l e t e l o w t e m p e r a t u r e

m e a s u r e m e n t s w o u l d b e n e c e s s a r y t o d e c i d e t h i s . A s i n N i , t h e l o w t e m p e r a t u r e

p T ) d a t a c a n n o t b e u n d e r s t o o d w i t h o u t i n c l u d in g th e p , ~ t e r m .

2.3.4. A llo ys containing interstitial impurities

N i , C o o r F e b a s e d a l lo y s c o n t a i n i n g s m a l l c o n c e n t r a t i o n s o f i n t er s t it i a l im p u r i t ie s

o f B o r C c a n b e p r e p a r e d b y r a p i d q u e n c h i n g . S w a r t z ( 19 71 ), S c h w e r e r ( 19 72 ) a n d

Ca dev i l l e a nd L e r n e r (1976) ha ve inv es t ig a ted the re s i s t iv i ty o f N iC , C__o_oC,

Nil_xF_eexC a l loys . Th e re s id ua l r e s i s t iv i ty o f th e se a l loys i s equ a l to ab ou t

3 . 4 ~ f l c m / a t . f o r C i n N i a n d 6. 6 t x O c m / a t . f o r C i n C o . F r o m t h e d e v i a t i o n s

f r o m M a t t h i e s s e n s ' r u l e i n N__iiCrC a n d C o C u C a l l o y s , C a d e v i l l e a n d L e r n e r ( 19 76 )

h a v e e s t i m a t e d t h a t t h e r e s i s t i v i t y P0+ w a s a b o u t t w i c e a s l a r g e a s Pot. T h i s r e s u l t ,

t o g e t h e r w i t h m a g n e t i z a t i o n a n d t h e r m o - e l e c t r i c d a t a b y t h e s a m e a u t h o r s , a r e

c o n s i s t e n t w i t h a p r e d o m i n a n t s c r e e n i n g b y t h e e l e c t r o n s o f th e d + b a n d . I n

N i~ -x F ex C a l l o y s t h e r e s i s t iv i t i e s P o t a n d p 0+ o f t h e C i m p u r i t i e s a r e f o u n d t o

b e c o m e n e a r l y e q u a l f o r x > 0 . 4 , w h i c h h a s b e e n a s c ri b e d t o t h e c h a n g e f r o m

s t r o n g t o w e a k f e r r o m a g n e t i s m ( C a d e v i l l e a n d L e r n e r 1 9 7 6 ) .

T h e r e s i st i v it y o f B i m p u r i t i e s in N i a n d C o h a v e b e e n f o u n d t o b e f a i rl y s m a l l

( - 1 i x l ) c m / a t . ) . T h i s h a s b e e n a s c r i b e d t o a p r e d o m i n a n t s c r e e n i n g b y t h e d ~,

e l e c t r o n s r e s u l t i n g in a s m a l l r e s is t i v i ty f o r t h e s p i n 1' e l e c t r o n s ( C a d e v i l l e a n d

L e r n e r 1 9 7 6 ) .




2.4. High temperature and critical point behaviour

I t was observed a long t ime ago tha t the res is t iv i t ie s of fe r romagne t ic meta ls

c ha nge d s lope a s a f unc t ion o f t e m pe r a tu r e a t t he C ur i e t e m pe r a tu r e . F o r N i th i s

was or ig ina l ly in te r pre te d by M ot t (1936) as indica t ing a reduc t i on of the sp in T

resist ivi ty on orde r ing. L at er work (Ka suy a 1956, Yo sh ida 1957, Coles 1958, We iss

and Marot ta 1959) showed tha t sp in d isorder sca t te r ing provided a more genera l

e xp la na t ion . Whe n the r e s i s t iv i t i e s o f the 3d f e r r om a gne t i c m e ta l s a r e c om pa r e d

wi th those o f the i r non- m a gne t i c 4d a n d 5d c o un te r pa r t s i t c a n be se e n c l e ar ly tha t

the re i s an ext ra magne t ic sca t te r ing cont r ibut ion which i s approximate ly cons tant

abo ve T~ and w hich decr eases grad ually belo w T~ (fig. 10) . Th e sim plest diso rder

mo de l shows tha t the pa ram agne t ic te rm above T~ is equa l to

kv mF)2 ttr

Pm= 4~e2zfi3 ~ + 1) ,

(26)

wh ere J i s the e f fec t ive loca l sp in and f the loca l sp in cond uc t ion e lec t ron sp in

coupl ing paramete r . De Gennes and Fr iede l (1958) sugges ted tha t the c r i t ica l

magn e t ic sca t te r ing near Tc was s imi la r in typ e to th e c r i t ica l sca t te r ing of

ne u t r on s a nd tha t i t shou ld l e a d to a pe a k in p T) a t To. La te r work by F isher and

I [

8

[3 Tc

cm ~

6C

c

Pd

20

~ I r T 80

0 I L I I

0 1 2 3 /~

F ig 10 Res i s t iv i ty of s eve ra l t r ans i t ion me ta l s a s a fun c t ion of

T/OD

OD i s t h e D e b y e t e m p e r a t u r e



774 I .A . C A M PB E L L A N D A . FE R T

Langer 1968), us ing a better approximation for the spin-spin correlat ion function

near To, mod ified this prediction to that of a peak in dp dT at To. They also made

the important remark that just abov e To the same leading term in the spin-sp in

corre la t ion should dominate dp dT and the magnetic specif ic heat , so that these

two parameters shou ld have the same crit ical behavio ur as T t ends to Tc from

above. Both magnetic entropy S and magnetic scatter ing rate should be propor-

t ional to

fo2kv F k, T) k 3 dk, (27)

where F k, T) is the spin-spin correlation function. Later theoretical work

showed that the same correspondence should hold equally in the region just

below Tc (Richard and Geldart 1973).

Renormalization theory can predict the critical coefficients for dp/dT (Fisher

and Aharony, 1973) but it is difficult to decide over what range of temperature

each side of To the strictly critical behaviour should be observed; Geldart and

Richard (1975) discussed the cross-over from a regime near To where the short-

range correlations dominate to a long-range correlation regime. The theory of

resistivity behaviour at To in weak ferromagnets has been developed by Ueda and

Moriya (1975), Der Ruenn Su and Wu (1975).

Experimentally, the critical behaviour of

dp/dT

has been studied very carefully

for Ni, Fe, Gd and the compound GdNi2 (Craig et al. 1967, Zumsteg and Parks

1970, Shaklette 1974, Kawatra et al. 1970, Zumsteg and Parks 1971, Parks 1972,

Zumsteg et al. 1970). For Ni (Zumsteg and Parks 1970) and Fe (Shaklette 1974) it

is found that dp/dT and the specific heat do indeed show the same A point type of

behaviour around To (fig. 11). The data are parameterized using

J i i

.0.05

.OOl

.005

.002 o ° ° 2

,001

o

o

o

.000

348 352 356

i l l ,

ooo o °

oo

o

o

o

oo

o

oo

oO o

o

o

o

o

o o

i

360 T(o C)36 368 372 376

1.04

1.03

1.02

1.01 -~

1,00 n,-

099

n*

0.98

0 97

096

Fig. 11. Resistivity

R T)

of nickel and

dR/dT

versus temperature in the region of the Curie point

after Zumsteg and Parks 1970).




1 dp A

p c d T - h ( e - * - l ) + B ,

T > To, (28)

a n d

1 d p A '

p o d T - - h

( le l- ~ '- 1 ) + B "

T < To, (29)

w h e r e

e = ( T - Tc)lTc. (30)

R e n o r m a l i z a t i o n t h e o r y p r e d i c ts h = h ' ~ 0 . 1 0 a n d

A/A ~ -1 . 3

(Zums teg e t a l .

1970) fo r a 3 d i me ns i ona l e xc ha nge f e r roma gne t . T he a c c u ra t e de t e rmi na t i on o f A

and h ' i s ex t rem ely de l i ca te espec ia l ly as Tc mus t b e f i t t ed se l f -cons i s t en t ly f ro m

t he da t a a nd i t a ppe a r s e s s e n t i a l t o ha ve t he t he o re t i c a l p r e d i c t i ons a s a gu i de .

In pu re F e , K ra f t m a k he r a nd P i ne g i na (1974) f ind h, h '= 0 -+ 0 .1 w h i le S ha k l e t t e

(1974) obs e rve s A , A '= -0 . 12 - -_ 0 . 0 1 by i mpos i ng h - - --h '. A gre e m e n t w i th t he

ma gne t i c spec i f ic hea t da ta in Fe i s ve r y good (Shak le t t e 1974, Co nne l y e t a l.

1971). Fo r Ni , the va lu es ob ta in ed we re h = 0 .1 _+0.1 , h ' = 0 .3_+0.1 (Zu ms teg and

Parks 1970) but wi th in the f i t t ing accuracy th i s i s presumably a l so cons i s t en t wi th

theore t i ca l va lues .

In G d w h i c h i s he xa gona l t he c r i t i c a l be ha v i ou r l ooks ve ry d i f f e r e n t w he n

me a s u re d a l ong t he c - a nd t he a - a xe s . Z ums t e g e t a l . ( 1970) s ugge s t t ha t t he

res i s t iv i ty changes a re compl ica ted by the c r i t i ca l behaviour of the l a t t i ce

pa ra me t e r s , bu t t h i s ha s be e n que s t i one d (G e l da r t a nd R i c ha rd 1975) .

G dN i 2 w a s i nve s t i ga t e d i n t he hope t ha t i t w ou l d c o r r e s pond t o a s i mp l e l oc a l

m o m e n t s y s t e m , dp/dT s how s s i mil a r cr it ic a l be ha v i ou r t o F e a nd N i bu t ha s m ore

c o m p l i c a t e d t e m p e r a t u r e d e p e n d e n c e a f e w d e g r e e s a b o v e Tc ( K a w a t r a e t al.

1970, Zums teg and Parks 1971) . The s igni f i cance of th i s has been d i scussed

(G e l da r t a nd R i c ha rd 1975) . T h e r e s i st i v it y va r i a t i on ha s a ls o be e n m e a s u re d a t

t he s t r uc t u r a l a nd f e r roma gne t i c t r a ns i t i on i n T bZ n (S ous a e t a l . 1979) .

T he c r i t i c a l be ha v i 0u r o f

dp/dH

ha s be e n s t ud i e d fo r N i (S c hw e re r 1974) a nd

fo r G d w i t h t he c u r r e n t i n t he ba s a l p l a ne (S i mons a nd S a l omon 1974) .

T h e b e ha v i ou r o f t r a ns po r t p rope r t i e s ne a r T c c a n a l s o be s t ud i e d in a l loys , bu t

loca l inh om og ene i ty l eads to a spread in the local va lues of Tc a t d i f fe rent pa r t s of

t he s a mpl e a nd s o t he c r i ti c al be ha v i o u r is s me a re d ou t . T h i s has be e n obs e rv e d i n

NiCu a l loys (Sousa e t a l . 1975) and in PdFe (Kawat ra e t a l . 1970, Kawat ra e t a l .

1969).

F i na l l y , be ha v i ou r a t t he c r i t i c a l c onc e n t r a t i on fo r f e r roma gne t i s m ( t he c onc e n -

t r a t i on a t w h ic h T o-> 0 ) c a n be s t ud i e d . V e ry va r i e d b e ha v i ou r ha s b e e n fou nd i n

di f fe rent a l loy sys tems . In N iCu a l loys the re i s a pea k in

dp/dT

at T~ as lon g as Tc

exi s t s and the re i s a maximum in

p(T)

som e de gre es high er , w hi le fo r c > cent a

m i n i m u m i n p(T) i s obs e rve d (H ough t on a nd S a ra c h i k 1970) . I n N i A u ( s p l a t

c oo l e d t o a vo i d s e g re ga t i on )

p(T)

shows a ma xi m um at T0 for c < cent (Ty ler e t a l .




1 9 73 ). F o r N i C r , Y a o e t a l. ( 1 97 5 ) f in d w e a k m i n i m a i n p T ) f o r c > c r i t w h i l e

S m i t h e t a l. fi n d g i a n t m i n i m a i n t h e r e g i o n c - c en t ( S m i t h e t al . 19 7 5 ). In N i P d

a l lo y s , T a r i a n d C o l e s ( 1 97 1 ) e x p r e s s t h e l o w t e m p e r a t u r e r e s i s t iv i t y b e h a v i o u r a s

p = p o = A T n a n d f i n d A is s h a r p l y p e a k e d a t c c~ t w h i l e n h a s a m i n i m u m w i t h

n - 1 . T h e C u r i e p o i n t i s n o t e a s y t o d e t e c t o n t h e p T ) c u r v e s . A m a m o u e t a l.

( 19 7 5 ) u s i n g t h e s a m e w a y o f e x p r e s s i n g t h e r e s is t iv i t y b e h a v i o u r f o u n d n --> 1 a n d

s t r o n g p e a k s i n A a t th e c r i ti c a l c o n c e n t r a t i o n s o f a l a r g e n u m b e r o f a ll o y s

s y s t e m s .

T h e t r a n s i t i o n f r o m l o w t e m p e r a t u r e t w o c u r r e n t b e h a v i o u r t o h i g h t e m -

p e r a t u r e s p i n d i s o r d e r b e h a v i o u r h a s b e e n s t u d i e d i n F e b a s e d a l l o y s ( S c h w e r e r

a n d C u d d y 1 97 0). T h e h i g h t e m p e r a t u r e r e si st iv i ty b e h a v i o u r o f th e a l lo y s e e m s t o

d e p e n d e s s e n t i a l l y o n t h e l o c a l i m p u r i t y m o m e n t .

3 . O t h e r tr a n s p o r t p r o p e r t i e s o f N i C o F e a n d t h e i r a l l o y s

H e r e w e w i l l s u m m a r i z e r e s u l t s o n d i f f e r e n t t r a n s p o r t p r o p e r t i e s i n t h e s e m e t a l s

a n d a l l o y s a n d o u t l i n e t h e i n t e r p r e t a t i o n s w h i c h h a v e b e e n g i v e n . W e w i l l

g e n e r a l l y f i n d t h a t N i h a s b e e n s t u d i e d i n m o s t d e t a i l w h i l e r a t h e r l e ss is k n o w n

a b o u t F e a n d C o . I n t h e i n t e r p r e t a t i o n o f t h e r e s u lt s , w e w i ll r e f e r t o w h a t h a s

b e e n l e a r n t a b o u t t h e d i f fe r e n t s y s te m s f r o m t h e r e s is t iv i ty m e a s u r e m e n t s w h i c h

w e h a v e a l r e a d y d i s c u s s e d .

3 .1 . O r d i n a r y m a g n e t o r e s i s t a n c e

W e h a v e o u t l i n e d t h e s i tu a t io n f o r p u r e m e t a ls i n se c t io n 2 .2 . F o r n o n - m a g n e t i c

a l l o y s t h e l o w t e m p e r a t u r e m a g n e t o r e s i s t a n c e b e h a v i o u r g e n e r a l l y f o l l o w s K o h -

l e r ' s r u l e ( p ( B ) - p ( O ) ) / p ( O ) = f B / p O ) ) , w h e r e f i s a f u n c t i o n w h i c h v a r i e s f r o m

m e t a l t o m e t a l b u t w h i c h i s r a t h e r i n s e n s i t i v e t o t h e t y p e o f i m p u r i t y p r e s e n t f o r a

g i v e n ho s t . I n a f e r r o m a g n e t a b o v e t e c h n ic a l s a t u ra t i o n t h e s a m e e f fe c t, d u e t o t h e

L o r e n t z f o r c e o n t h e e l e c t r o n s , ca n b e o b s e r v e d b u t a s B i n c lu d e s t h e m a g -

n e t i z a t i o n t e r m 4 7 r M , p ( 0 ) c a n n o t b e a t t a i n e d e x c e p t b y e x t r a p o l a t i o n . S c h w e r e r

a n d S i lc o x ( 19 7 0 ) s h o w e d b y a c a r e f u l s t u d y o f d i lu t e N i a l l o y s a m p l e s t h a t f o r a

g i v e n s e r i e s o f a l l o y s ( e . g . N i F e s a m p l e s ) t h e o r d i n a r y m a g n e t o r e s i s t a n c e f o l l o w s a

K o h l e r ' s r u le , b u t t h a t t h e K o h l e r f u n c t i o n f v a r i e d c o n s i d e r a b l y w i th t h e t y p e o f

s c a t t e r e r ( fig . 1 2). O t h e r w o r k ( F e r t e t a l . 1 97 0 , D o r l e i j n 1 9 7 6 ) i s c o n s i s t e n t w i t h

t h e s e d a t a .

I t c a n b e s e e n i n fig . 1 2 t h a t t h e s t r o n g e s t m a g n e t o r e s i s t a n c e s a r e a s s o c i a t e d

w i t h i m pu r i t i e s ha v i n g l a r ge va l ue s o f p0+ / P0 t ( i .e . N__iiFe, __Ni Co . . . ). T h e l on -

g i t u d i n a l m a g n e t o r e s i s t a n c e o f t h e s e a l lo y s is a l so h i g h [Apll /P O ) s a t u r a t e s a t a b o u t

1 0 i n N i F e ( S c h w e r e r a n d S i l c o x 1 9 7 0 ) ] c o n s i d e r a b l y g r e a t e r t h a n t h a t o b s e r v e d

f o r C u b a s e d a ll o y s f o r i n st a n c e , w h e r e

A p J p O )

s a t u r a t e s a t a b o u t 0 . 7 ( C l a r k a n d

P o w e l l 1 9 6 8 ) . A t t e m p t s h a v e b e e n m a d e t o u n d e r s t a n d t h i s b e h a v i o u r i n t h e t w o

c u r r e n t m o d e l . I n i ts s i m p l e s t f o r m t h e t w o t y p e s o f e l e c t r o n ( sp i n 1' a n d s p in $ )

c a n b e r e p r e s e n t e d b y e l e c t r o n - l ik e s p h e r e s i n k s p a c e w i th d i f fe r e n t r e l a x a t io n



TRANSPORT PROPERTIES OF FERROMA GNETS 777

. o o o / / / /

1.05

I D O ~ ~ R u ~

0 10 20 30 O 50

B/~O (k G/,u.~.cm)

Fig. 12. Ko hler plots for the transverse m agnetoresistance at 4.2 K of nickel containing Co, Fe, M n,

Ti, A1, Cr, Pt, V o r R u im purities (after Dorleijn 1976).

t i m e s . I n t h i s a p p r o x i m a t i o n ( F e r t e t a l . 1 9 7 0 ) t h e t r a n s v e r s e m a g n e t o r e s i s t a n c e i s

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

p~ O)/p t

( 0) , b u t t h e m o d e l i s n o t s a t i s f a c t o r y a s i t

p r e d i c t s a z e r o l o n g i tu d i n a l m a g n e t o r e s i s t a n c e in d i s a g r e e m e n t w i t h e x p e r i m e n t .

A s a n e x t s t e p , it i s p o s s i b l e t o i n v o k e r e l a x a t i o n t i m e a n i s o t r o p y w i t h i n e a c h s p in

b a n d . D o r l e i j n ( 1 97 6 ) s u g g e s t s t h a t t h e in t r in s i c m a g n e t o r e s i s t i v i t y o f t h e s p in 1

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

t r a n s v e r s e m a g n e t o r e s i s t a n c e s a r e m u c h g r e a t e r w h e n t h e c u r r e n t i s c a r r ie d

m a i n l y b y t h e s p in 1 e l e c t r o n s . J a o u l ( 1 9 74 ) p r o p o s e s t h a t t h e r e is a m i x i n g

b e t w e e n s p in 1 a n d s pi n ~ c u r r e n t s w h i c h is a n i n c r e a s in g f u n c t io n o f

B/p O).

T h i s is b e c a u s e s p i n - o r b i t e f f ec t s m e a n t h a t a n e l e c t r o n o n a g iv e n o r b i t o n t h e

F e r m i s u r f a c e p a s s e s c o n t i n u o u s l y b e t w e e n s p i n 1 a n d s p in + , p r o g r e s s i v e l y

m i x i n g c u r r e n t s a s

B / p O )

i n c r e a s e s . T h i s m o d e l p r e d i c t s t h e s a t u r a t i o n m a g -

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

T h e o r d i n a r y m a g n e t o r e s i s t a n c e i n F e b a s e d a l lo y s i s m o r e d if fi cu lt to e x p r e s s i n

t h e f o r m o f K o h l e r c u rv e s , b e c a u s e t h e m u c h h i g h e r v a lu e o f 4 ~ M in F e m e a n s

t h a t e x t r a p o l a t i o n s t o B = 0 a re a l w a y s v e r y e x t e n d e d . D a t a g i v e n b y D o r l e i j n

( 19 7 6) a g a i n i n d i c a t e d i f f e r e n t K o h l e r c u r v e s f o r F e s a m p l e s c o n t a i n i n g d i f f e re n t

i m p u r i t i e s , b u t t h e c o r r e l a t i o n w i t h t h e v a l u e o f

p+ O)/p,

( 0) i s m u c h l e ss c l e a r t h a n

i n t h e c a s e o f N i b a s e d a l l o y s .

T h e r e is a n a d d i t i o n a l e f f ec t t h a t a p p e a r s u n d e r s i m i l a r e x p e r i m e n t a l c o n d i t i o n s

a s t h e L o r e n t z f o r c e o r d i n a r y m a g n e t o r e s i s t a n c e , b u t w h i c h i s d u e t o t h e h ig h f ie l d

s u s c e p t i b i l i t y o f t h e f e r r o m a g n e t i c m e t a l . T h i s h i g h f i e l d s u s c e p t i b i l i t y c a n h a v e

t w o o r i g in s . F i rs t , t h e r e i s a n i n c r e a s i n g m a g n e t i c o r d e r i n a n a p p l i e d f i e l d w h i c h

c a n a l so b e t h o u g h t o f a s a r e d u c t i o n in t h e n u m b e r o f m a g n o n s w i t h i n c r e a s in g

f ie ld . T h i s t e r m is m a x i m u m a r o u n d T c a n d g o e s t o z e r o a s T g o e s t o z e r o .

S e c o n d l y , fo r a b a n d f e r r o m a g n e t , t h e l oc a l m a g n e t i c m o m e n t s c a n b e a l t e r e d b y

a n a p p l i e d f ie ld a t a n y t e m p e r a t u r e , e v e n T = 0 ( V a n E l s t 1 95 9).




I ns o f a r , a s a n i nc r e a s i ng f i e l d p r oduc e s i nc r e a s i ng ma gne t i c o r de r a nd he nc e

lower sp in d i sorder sca t t e r ing ,

d p / d H

due to the f i r s t t e rm wi l l be nega t ive . The

second type of e f fec t can in pr inc ip le g ive e i the r pos i t ive or nega t ive mag-

ne t o r e s i s t a nc e de pe nd i ng on t he e l e c t r on i c s t r uc t u r e o f t he s y s t e m. V a n E l s t

(1959) meas ure d a t 300 K

1/p) dp/dH) l I ~- 1 /p ) dp /dH ) l

w i t h e f f e c ts o f t he o r de r

of 10-4 /kG and wi th s ign i f i can t va r i a t ions f rom one a l loy to another . Thi s

be ha v i ou r i s due t o t he f i r s t e f f e c t . A t l ow t e mpe r a t u r e s t he L o r e n t z - f o r c e

ma g ne t o r e s i s t a nc e dom i na t e d e xc e p t f o r N i M n a l loys w h i c h s how e d ne ga t i ve

d p / d H e ve n a t l ow t e mpe r a t u r e ; t h i s i s p r oba b l y due t o a n unus ua l ba nd

suscept ibi l i ty in these a l loys .

3.2. Ordin ary H al l coefficient

I n non - ma g ne t i c m e t a l s i t is know n t ha t t he o r d i na r y H a l l c oe f f ic i e n t R 0 be ha v e s

t o a r ough a pp r ox i ma t i on a s

Ro oc 1/e n*

wh ere n* i s the e f fec t ive dens i ty of

cur ren t ca r r i e r s and e i s the i r charge (e i s nega t ive for e l ec t ron- l ike ca r r i e r s and

pos i t ive for ho le - l ike ca r r i e r s ) . The ac tua l va lues of R0 can be cons ide rab ly

mod i f i e d by F e r mi s u r f a c e a nd s c a t t e r i ng a n i s o t r opy e f f e c t s ( H ur d 1972) ; f o r t he

high f i e ld cond i t ion wc >> 1 , R0 dep end s only on the Ferm i sur face g eo m et ry and

can be h ighly an i so t ropic in s ing le c rys ta l s .

I n f e r r oma gne t i c me t a l s t he o r d i na r y H a l l ef f e c t c a n be s e pa r a t e d f r om t he

e x t r a o r d i na r y H a l l e f fe c t by me a s u r e m e n t s a bov e t e c hn i ca l s a t u r a t ion , a s l ong a s

the suscept ib i l i ty of the sample in h igh f i e lds i s negl ig ib le so tha t the re i s no

pa r a m a gne t i c e x t r a o r d i na r y H a l l e f f e c t c o r r e c t i on ( se e s e ct i on 3 .4 ).

T h e o r d i na r y H a l l c oe ff i c ie n t i n Ni a t r oom t e m pe r a t u r e i s R 0-~

-6 x 10 1312cm/G (Lavine 1961) , which cor responds to conduc t ion by e lec t ron-

l i ke c a r r i e rs w i t h a n e f f e c t i ve e l e c t r on de ns i t y n* o f a bou t 1 e l e c t r on pe r a t om. R 0

v a r ie s b y a b o u t 20 b e t w e e n r o o m t e m p e r a t u r e a n d 5 0 K ; a t l o w e r t e m p e r a t u r e s

the low f ie ld con di t io n ~0c~-'~ 1 no lon ger hold s fo r high p uri ty Ni sa mp les so R0

t e nds t ow a r ds t he h i gh f i el d va l ue ( R e e d a nd F a w c e t t 1964) .

Pugh and coworkers (Pugh e t a l . 1955 , Sandford e t a l . 1961 , Ehr l i ch e t a l . 1964)

a nd S mi t ( 1955) s how e d t ha t f o r a numbe r o f N i ba s e d a l l oys , i n pa r t i c u l a r N i F e

a nd N i C u , t he l ow t e mp e r a t u r e H a l l c oe ff i c ie n t s i n c on c e n t r a t e d s a mpl e s c o r -

r e s p o n d t o m u c h l o w e r e ff e c ti v e c a r r ie r co n c e n t r a ti o n s , n * - 0 . 3 e l e c tr o n s p e r

a t om. T he y po i n t e d ou t t ha t t h is l ow num be r o f c a rr i e r s w a s p r oba b l y , a s s oc i at e d

w i t h a r e g i me w he r e on l y t he c onduc t i on ba nd f o r one d i r e c t i on o f s p i n w a s

c a r r y i ng t he c u r r e n t . L a t e r w or k on N i a nd N i C u a l l oys ( D u t t a R oy a nd S ubr a h -

ma ny a m 1969) s how e d t ha t R 0 is ve r y t e m pe r a t u r e d e pe nd e n t i n t he a l loys , a nd

t ha t a b ove t he C ur i e po i n t n* r e t u r ns t o a va l ue o f a bou t 1 e l e c t r on - a t om , i .e ., t o

a s i t ua t i on w he r e bo t h s p i n d i r e c t i ons c a r r y c u r r e n t .

T h i s w ou l d s e e m t o f i t i n w e l l w i t h o t he r da t a on t he t w o c u r r e n t mode l .

H o w e v e r , c a r e f u l m e a s u r e m e n t s b y H u g u e n i n a n d R i v i e r ( 1 9 6 5 ) a n d b y M i e d e m a

a nd D or l e i j n ( 1977) on a w i de r a nge o f N i ba s e d a l l oys ha ve s how n t ha t t he

s i t ua t i on i s mor e c ompl i c a t e d . T he da t a c a n be s umma r i z e d a s f o l l ow s : t he l ow

t e m pe r a t u r e R 0 is ve r y c l o se t o z e r o in d i l u t e a l loys ( c onc e n t r a t i on - 0 . 5 ) f o r




w h i c h Po+/Pot > 1 ( i. e. N i F e , N __iiCu, N i C o . . . ) b u t t h e n i n c r e a s e s r a p i d l y w i t h

i m p u r i t y c o n c e n t r a t i o n t o a v a l u e c o r r e s p o n d i n g t o n * - 0 .3 i n s a m p l e s w h e r e t h e

i m p u r i t y r e s i s t i v i ty is g r e a t e r t h a n a b o u t 5 tx f~ cm . F o r a l l o y s f o r w h i c h

po ~/po t < 1,

R 0 i s e s s e n t i a ll y i n d e p e n d e n t o f i m p u r i t y c o n c e n t r a t i o n a t a b o u t - 6 x 1 0 -13 l ~ c m / G

( n o t e t h a t o n l y s a m p l e s o f t h is t y p e h a v i n g p > 2 i x ~ c m w e r e s t u d i e d ) .

N o w i n a t w o c u r r e n t m o d e l R 0 i s g i v e n b y

Ro = p2 Ro t /p2 + Ro+/p~ ,

31 )

w h e r e R o t , R 0+ a r e t h e o r d i n a r y H a l l c o e f f ic i e n ts f o r t h e t w o s p i n d i r e c ti o n s

t a k e n s e p a r a t e l y . F r o m t h e e x p e r i m e n t a l d a t a i t c a n b e c o n c l u d e d t h a t R 0 + i s

r e a s o n a b l y c o n s t a n t , w h il e R o t v a r ie s s t ro n g l y w i t h p ~ . D o r l e i j n a n d M i e d e m a

s u g g e s t e d t h a t t h e e f f ec t i s d u e t o a s m u d g i n g o u t o f t h e d e t a il s o f t h e s p in 1'

F e r m i s u r f a c e o f N i w i t h i n c r e a s i n g P t a n d t h e y a s s o c i a t e d t h i s w i t h t h e o b s e r v e d

c h a n g e s o f t h e m a g n e t o c r y s t a l l i n e a n i s o t r o p y w i t h a l l o y c o n c e n t r a t i o n ( M i e d e m a

a n d D o r l e i j n 1 97 7) . A s w e w i ll s e e i n s e c t i o n 3 .3 , t h e r e s i s t i v it y a n i s o t r o p y o f t h e

s a m e a l lo y s c h a n g e s s i m i l a rl y w i t h i m p u r i t y c o n c e n t r a t i o n u n t i l a c e rt a i n r e s i d u a l

r e si s ti v it y v a l u e is r e a c h e d . T h e R 0 d a t a s ug g e s t t h a t th e s m u d g e d o u t F e r m i

s u r fa c e s i tu a t io n c o r r e s p o n d s m o r e c l os e ly to t h e e x t r e m e s - d m o d e l w i t h c o n-

d u c t i o n e n t i r e l y b y a n s t l ik e b a n d c o n t a i n i n g a b o u t 0 . 3 e l e c t r o n s p e r a t o m .

T h e r e s u l t s o n R 0 i n F e b a s e d a l lo y s a r e l e ss c le a r , p a r t ly b e c a u s e t h e s e p a r a t i o n

i n t o o r d i n a r y a n d e x t r a o r d i n a r y H a l l c o m p o n e n t s i s m o r e d if fi cu lt b e c a u s e o f th e

l a r g e v a l u e o f 4 ~ - M . F e h a s a p o s i t i v e o r d i n a r y H a l l c o e f f i c i e n t , a s h a v e t h e d i l u t e

F e b a s e d a l l o y s e x c e p t f o r F e C o ( B e i t e l a n d P u g h 1 95 8) a l t h o u g h R 0 f o r __FeNi

a l lo y s c h a n g e s s i g n w i t h t e m p e r a t u r e a n d w i t h c o n c e n t r a t i o n ( S o f te r e t a l. 1 9 65 ).

T h e r e a p p e a r s t o b e e v i d e n c e ( C a r t e r a n d P u g h 1 96 6) th a t a l l o y s f o r w h i c h

p t O) /p+ O)> 1 s u c h a s F e C r , b e h a v e s i m i l a r l y t o N i i n t h a t R 0 is h i g h a t l o w

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

t i o n s b e g i n t o p a r t i c i p a t e i n t h e c o n d u c t i o n .

3.3. Spontaneous resistivity anisotropy

T h i s w a s d e f i n e d i n s e c t i o n 1 a n d i s a s p i n o r b i t e f f e c t . T h e m e c h a n i s m c a n v a r y

f r o m s y s t e m t o s y s t e m . T h e s i m p l e s t c a s e t o u n d e r s t a n d , a t l e a s t i n p r i n c i p l e , i s

t h a t o f d i l u t e r a r e e a r t h i m p u r i t i e s ( F e r t e t a l. 19 77 ). B e c a u s e o f t h e u n c l o s e d f

s h e ll , t h e m a g n e t i c r a r e e a r t h s c a n b e r e g a r d e d a s io n - l i k e w i t h a n o n - s p h e r i c a l

d i s t r i b u t i o n o f c h a r g e ( a p a r t f r o m t h e s p h e r i c a l i o n G d 3 + ) . A c o n d u c t i o n e l e c t r o n

p l a n e w a v e e n c o u n t e r s a n o b j e c t w i t h a d i f f e r e n t c r o ss s e c ti o n d e p e n d i n g o n

w h e t h e r i t a r r i v e s w i t h i t s k v e c t o r p a r a l l e l o r p e r p e n d i c u l a r t o t h e r a r e e a r t h

m o m e n t , w h i c h p r o v i d e s a n a x i s f o r t h e n o n - s p h e r i c a l c h a r g e d i s t r i b u t i o n . T h e

a n i s o t r o p y o f t h e r e s i st iv i ty is p r o p o r t i o n a l t o t h e e l e c t ro n i c q u a d r u p o l e m o m e n t o f

t h e p a r t i c u l a r r a r e e a r t h . T h e t h e o r y o f t h is e ff e c t h a s b e e n w o r k e d o u t i n d e t a i l (F e r t

et a l . 1977).

I n t r a n s i t i o n m e t a l s , t h e s p i n o r b i t c o u p l i n g i s u s u a l l y a w e a k p e r t u r b a t i o n o n

t h e s p i n m a g n e t i z a t i o n . T h e l o w e s t o r d e r t e r m s l e a d i n g t o a r e s i st i v it y a n i s o t r o p y



780 I.A. CAM PBELLAND A. FERT

wil l be e i the r mixing te rms of the type L + S - ) 2 o r pola r iza t ion te rms of the type

(LzSz)2. Smit (1951) calcula ted the resist ivi ty anisotropy to be expected on an s-d

mo de l f rom the mixing te rms ac t ing be tw een sp in 1' and sp in ,~ d bands . W hen

da ta be c a m e a va i l a b le f o r bo th the a n i so t r opy a nd the

p ~/p t

ra t ios in va r ious Ni

a l loys , i t wa s f ound tha t t he r e wa s good a g r e e m e n t be twe e n the r e su l t s a nd

predic t ions which could be made us ing the Smit approach (Campbe l l e t a l . 1970) .

Ag r e e m e n t i s how e ve r l e ss good f o r im pur i ti e s ha v ing a vi r tua l boun d d s t a t e ne a r

the Ferm i sur face , and an addi t iona l (LzS~) 2 mec hanis m was sugges ted for these

cases (Jaoul e t a l . 1977 .

Th e re la t ive an iso t rop y of the res is t iv i ty (P ll - P± /P def ined in sec tion 1 has b een

measured for Ni and a la rge number of Ni a l loys as a func t ion of concent ra t ion

and tempera ture (Smit 1951, Van Els t 1959, Berger and Fr iedberg 1968, Campbe l l

e t a l . 1970, Vasi lyev 1970, Campbell 1974, Dedi6 1975, Dorle i jn 1976, Dorle i jn

and Miedema 1976, Kaul 1977, Jaoul e t a l . 1977) and for many d i lu te Fe based

a l loys , ma in ly a t He tempera ture (Dor le i jn and Miedema 1976) . We wi l l f i r s t

d iscuss the Ni da ta .

The a n i so t r opy r a t io f o r pu r e N i i s ne a r +2% f r om n i t r oge n t e m pe r a tu r e up to

r oom te m pe r a tu r e , a nd the n g r a dua l ly d rops a s t he t e m p e r a tu r e i s i nc r e a se d up to

the Curie point (Smit 1951, Van Elst 1959, Kaul 1977) . Below nitrogen tem-

pera ture the an iso t rop y is d i ff icu l t to e s t ima te for pure sam ples because of the

r a p id ly inc r e as ing o r d ina r y m a gne to r e s i s t a nc e , bu t i t a ppe ar s t o r e m a in f a ir ly

cons tant .

For most dilute N__iiXa l loy se r ies the l imi t ing low tem per a tu re an iso t ropy ra t io i s

re la t ive ly concent ra t ion independent for a g iven type of impur i ty X over a fa i r ly

wide c onc e n t r a t ion r a nge bu t t he va lue de pe nds s t r ong ly on the type o f im pur i ty ,

tab le 4. F or N iCo , N iF e an d N__iiCu (fig. 13) the an iso tro py r atio incre ases

c on t inuous ly w i th c onc e n t r a t ion up to c onc e n t r a t ions c o r r e spond ing to r e s idua l

res is t iv i t ie s of about 2 p~cm. I t i s a d isputed poin t a s to whe ther the appropr ia te

charac te r i s t ic va lue of the an iso t ropy ra t io for these a l loys i s the p la teau va lue

(Jaoul e t a l . 1977) or a va lue a t some lower concent ra t ion (Dor le i jn and Miedema

1975b, 1976).

Wh e n the t e m pe r a tu r e i s i nc r ea se d , t he a n i so t r opy r a t io o f a g ive n sa m ple t e nds

towards the pure Ni va lue and f ina l ly becom es ze ro a t the Cur ie poin t of the a l loy

(Vasilyev 1970, Kaul 1977).

The r e is a cl e a r c o r r e l a tion be tw e e n the va lue o f a a nd the low t e m pe r a t u r e

an is o t r o p y ra t io (Camp be l l e t a l . 1970). Al loy s having h igh va lues of a N(~Co,

NiFe , N i M n . . . ) have h igh pos i t ive res is t iv i ty an iso t ropies whi le a l loys wi th c~ ~ 1

have smal l pos i t ive or nega t ive an iso t ropies . A sp in-orb i t mixing mode l or ig ina l ly

sugges ted by Smit (1951) g ives a convinc ing explan a t ion of the overa l l va r ia t ion of

the an iso t ropy ra t io wi th the va lue of a . As Ni meta l has a fu l ly pola r ized d band,

the re a re no d 1' s ta tes a t the Ferm i su r face for the cond uc t ion e lec t rons to be

sc a t t e r e d to . Howe ve r be c a use o f t he sp in - o r b i t m ix ing by the m a t r ix e l e m e n t

AL+S- some d 1' cha rac te r is mixed in to the d band. The resu l t ing wea k s ]' to

d sca t te r ing can be shown to dep end s t rongly on the re la t ive or ien ta t ion of the k

vec tor of the s conduc t ion e lec t ron and the sample magne t iza t ion . This leads to a

res is t iv i ty an iso t ropy of the form




T A B L E 4

An iso t rop y of the res idu a l re s i s t iv ity of d i lu te n icke l based a l loys* .

781

I m p u r i ty C o F e M n C r V P d

PJ -

P±

x 102 20 Ca) 13 .6 o ' ) 9 . 9 ° ) - -0 .3 500) 0 . 6 C") 2 C~)

t~

14 .8 Cb) 14 Cd) 7 . 8 Cb) - - 0 . 28 (a) O. 15 c° )

28 (c) 19.5 Cc) 9.5 Co) - 0 .2 3 Cd)

I m p u r i ty R h R u M o N b P t I r

P_JI- P± x 10 2 0 .0 5 C°) - 0 .6 00 ' ) 0 .1 Ca) 0 . 15 (~ ) 0 . 4 C°) - 1 .5 2 C~)

0. 05 C~) -0 .8 2 C~ 0 . 05 (0) 0 . 4 C~

Impur i t y R e W C u Au A1 S i Zn S n

p p - P a x 102 -0 .5 0 C°) 0 . 4 Ca) 6 . 8 C°) 7 . 5 C°) 4 . 7 (a) 2 . 5 ( " ) 5 . 7 Ca) 3 . 4 Ca)

-0 .4 5 Cc 0 . 8 ¢ ) 7 . 8 Co) 7 . 9 Co) 3 . 9 Co 2 . 1 co 4 . .7 b) 2 . 9 Co

4. 6 Co) 2 . 8 (c)

6.5 ¢)

3 .5 Co)

*A ft er : ca)Van Elst 1959, C°)Dorleijn and M ied em a 1974, Dorlei jn 1976, e)J aou l e t a l. 1977,

ca) Sch we rer and Silcox 1970. W e indi cate - w he n this is possib le - t he resist ivi ty aniso trop y o f a l loys in

t he c onc e n t ra t i on r a nge wh e re t he c onc e n t ra t i on d e pe nd e nc e i s we a k ( s e e f ig . 13 ). The e xpe r i m e n t a l

da ta on (P / I -

P± /Pll

ha s be e n re -e xp re s se d i n t e rms o f (P H-

P± /P.

a tf

( )

C

2 5

2 0

5

10

5

0

o o o o o

i

\

x ~ \

x %,

i x L

io 4 6

I m p u r i t y c o n c e n t r a t o n , a t °/o

Fig . 13. Con cent r a t ion dep end enc e of the res i st iv i ty anisot rop y a t 4 .2 K for severa l n icke l based a l loys .

A A : Ni Co , OC): N iFe, ×: N_iiCu (aft er Jao ul et al. 1977).




P l l - P . / P = ~ o ~ - 1 ) ,

(32)

w h e r e y i s a s p i n - o r b i t c o n s t a n t w h i c h c a n b e e s t im a t e d t o b e a b o u t 0 . 0 1 f r o m t h e

N i g f a c t o r . T h i s m o d e l e x p l a i n s t h e s i g n , t h e m a g n i t u d e a n d t h e g e n e r a l v a r i a t i o n

o f t h e a n i s o t r o p y w i t h a (fig . 1 4). In a d d i t i o n , i t h a s b e e n s h o w n ( E h r l i c h e t a l.

1 9 64 , D o r l e i j n a n d M i e d e m a 1 97 6 , J a o u l e t a l. 1 9 7 7) t h a t a n a n a l y s is o f t h e

a n i s o t r o p y r a t i o o f t e r n a r y a l l o y s c a n l e a d t o e s t i m a t e s o f t h e i n d i v i d u a l a n i s o t r o -

p i e s f o r t h e sp i n 1' a n d s p i n c u r r e n t s a n d t h a t f o r a l l o y s w i t h a > 1 t h e r e s u l t s

a r e in a g r e e m e n t w i th t h e p r e d i c t i o n s o f t h e S m i t m e c h a n i s m .

H o w e v e r , f o r a n u m b e r o f a l lo y s o f N i f o r w h i c h c~ < 1 , a l t h o u g h t h e r e s is t iv i t y

a n i s o t ro p i e s r e m a i n s m a ll a s w o u l d b e e x p e c t e d f r o m t h e S m i t m e c h a n i s m , e q .

(3 2) is n o t a c c u r a t e l y o b e y e d a n d t h e a n i s o t r o p i e s o f t h e t w o s p i n c u r r e n t s d o n o t

o b e y t h e S m i t r u l e s ( E h r l i c h e t a t. 1 9 55 , J a o u l e t a l. 1 9 77 ). A f u r t h e r m e c h a n i s m

n e e d s t o b e i n v o k e d f o r th e s e s y s te m s , w h i c h a r e c h a r a c t e r i z e d b y v ir t u al b o u n d

s t a te s a t t h e s p in I' F e r m i l e v e l. A m e c h a n i s m h a s b e e n p r o p o s e d i n v o l v i n g t h e

) t L z S z s p i n - o r b i t i n t e r a c t i o n o n t h e i m p u r i t y s i t e , p a r t i c u l a r l y f o r i m p u r i t i e s w h i c h

h a v e s t r o n g s p i n - o r b i t i n t e r a c t i o n s ( J a o u l e t a l . 1 9 7 7 ) . D o r l e i j n a n d M i e d e m a

( 1 9 7 6 ) p o i n t e d o u t t h a t f o r m o s t i m p u r i t i e s , w h a t e v e r t h e v a l u e o f c ~ , ( A p / p ) t > 1

a n d

(Ap/p)

< 1 b u t t h e y d i d n o t e x p l a i n t h is r e g u l a r i ty .

T h e t e m p e r a t u r e v a r i a t i o n o f t h e a n i s o t r o p y r a t io c a n a ls o b e u n d e r s t o o d u s in g

t h e S m i t m o d e l ( C a m p b e l l e t a l . 1 9 7 0 ) . A s p h o n o n a n d m a g n o n s c a t t e r i n g

i n c r e a s e s w i t h i n c r e a si n g t e m p e r a t u r e , t h e e f f e c t i v e v a l u e o f a f o r a n a l l o y t e n d s

t o a p p r o a c h t h e p u r e m e t a l v al u e . D a t a o n N i C u a ll o y s h a v e b e e n a n a l y z e d in th i s

3

2

I

1o 2

3 0

Fig. 14. Resistivity anisotropy of Ni b ase d alloys at 4.2 K as a function of a = P0J,/P0t- The straight

line is Ap/~ = 0.01 (a - 1) (after Jao ul et al. 1977).




w a y ove r a w i de t e m pe r a t u r e a nd c on c e n t r a t i o n r a nge ( K a u l 1977 ) s o as t o e s t i m a t e

pit T), pi~ T) and p~ ~ (T) .

H i gh c onc e n t r a t i on e f f e ct s i n c e rt a i n a l loy s er ie s ha ve be e n i n t e r p r e t e d a s due

t o c ha r a c t e r i s t i c c ha nge s i n t he e l e c t r on i c s t r uc t u r e w i t h c onc e n t r a t i on ( C a m pbe l l

1974).

T he r e s i s t i v i t y a n i s o t r opy o f a l a r ge num be r o f F e ba s e d a l l oys ha s a l s o be e n

s t ud i e d ( D o r l e i j n a nd M i e de m a 1976) . H e r e , t he a l l oys ha v i ng

p, O)/p+ O),> 1

have s t rong pos i t ive res i s t iv i ty an i so t ropies whi le those wi th

Pl O)/p+

(0) < 1 ha ve

smal l an i so t ropy ra t ios ( t ab le 4) . Aga in , an ana lys i s in t e rms of the an i so t ropies of

t he s p in ]' a nd sp i n c u r r e n t s ha s be e n c a r r ie d ou t a nd t he p r e d i c t i ons o f t he

S m i t a pp r oa c h s e e m w e l l bo r ne ou t ( D or l e i j n 1976 ) .

A s w e ha ve s e e n i n s e c t i on 1 t he r e s i s t i v i t y a n i s o t r opy i n c ub i c f e r r om a gne t i c

m onoc r ys t a l s c a n be e xpa nde d i n a s e r i e s o f D 6r i ng c oe f f i c i e n t s k l . . . k s . O nc e

aga in , Ni and Ni a l loys have been the mos t s tud ied [pure Ni (Bozor th 1951) , Ni

15% Fe (B er ge r an d F ri ed be rg 1968), Ni 1.6% C r a nd N_j 3% F e (Jao ul 1974), N__ii

0 .5% Fe , Ni 0 .55% P t and Ni 4% Pd (Dedi6 1975) ] . Very roughly the ind iv idua l k i

c oe f f ic i e n ts a r e s i m p l y p r opo r t i ona l t o t he a ve r a ge po l yc r ys t a l a n i s o t r opy w i t h t he

e xc e p t i on o f k3 ( t a b le 5 ) . T h i s c oe f f i c ie n t m a y be ha ve d i f f e r en t l y f r om t he o t he r s

b e c a u s e i t d o e s n o t s tr ic tl y r e p r e s e n t a n a n i s o t r o p y - i t c o r r e s p o n d s t o a n a v e r a g e

c ha nge o f the s a m p l e r e s i st i v it y w i t h t he m o m e n t d i r e c t i on w h ic h is i nd e pe n de n t

o f t he c u r r e n t d i r e c t i on .

TABLE 5

Magnetoresistance anisotropy in Ni and Ni alloy single crystals. D6ring

coeff ic ients

ki are given n percent. References: (a~D6ring 1938, Co)Berger and

Friedberg 1968, (c~Ja oul 1974, cd~Dedi6 1975.

kl ke k3 k4 k5

Ni, 300 K (a) (d) -3.4 -5.2 + 1.7

NiFe 15% 4.2 K b~ 55.0 14.5 -26.3 -37 .8 +24 .7

NiCr 1% 4,2K c~ -3.0 -0.3 -1.2 + 2. 3 0_+0 .7

NiPd 4% 4.2 K d~ 4.0 1.0 -5 .5 -4 .0 -3 .0

T h e r e a r e a ls o m e a s u r e m e n t s o f t he D 6r i ng c oe f fi c ie n t s f o r F e a t r oom

t e m pe r a t u r e ( B oz o r t h 1951 ) .

N o c o n v in c i n g m o d e l h a s b e e n p r o p o s e d t o e x p la i n t h e m o n o c r y s t a l a n i s o t r o p y

c oe f fi c ie n t s w h i c h p r e s u m a b l y de p e nd on t he de t a i l e d ba nd s t r uc t u r e o f t he m e t a l .

T he f a c t t ha t t he t e r m s w h i c h a r e f ou r t h o r de r i n t he d i r e c t i on c os i ne s o f t he

magnetization k3, k4,

k5) are as large as the sec on d o rd er term s (kl , k2) i s

r e m a r k a b l e .

3.4. Extraordinary Hall effect

A pa r t f r om t he r e s i s t i v i t y , t he p r ope r t y o f f e r r om a gne t i c m e t a l s w h i c h ha s

a t t r a c t e d t he g r e a t e s t t h e o r e t i c a l a t t e n t i on is t he e x t r a o r d i na r y H a l l e f fe c t, R s; t he

e x t r a o r d i na r y H a l l vo l t a ge i s r e m a r ka b l e i n be i ng bo t h s t r ong a nd r a p i d l y va r y i ng




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

w hi c h a r e be l i e ve d t o p r od uc e t h is e f f e c t w e r e p r o pos e d s om e ye a r s a go by S mi t

(1955) and L ut t in ger (1958) bu t the phys ica l unders tand ing of these e f fec t s has

be e n c ons i de r a b l y i mpr ove d qu i t e r e c e n t l y ( B e r ge r 1970 , L yo a nd H o l s t e i n 1972 ,

N oz i b r e s a nd L e w i n e r 1973) . W e w i ll ou t l i ne t he d i s c uss i on g i ve n by N o z i b r e s a nd

L e w i ne r ( 1973) ; a l though t h is t he o r y w a s d e ve l o pe d s pec if ic a ll y f o r s e mi c ondu c -

t o r s t he s a me phys i c s c a n b r oa d l y be u s e d f o r f e r r oma gne t i c me t a l s .

A n e l e c t r on i n a ba nd s ubmi t t e d t o t he s p i n - o r b i t i n t e r a c t i on a c qu i r e s a n

e f f e c ti ve e l e c tr i c d i po l e m om e n t

p = - A k x s ,

w h e r e A is a s p i n - o r b i t pa r a m e t e r , k is t he k ve c t o r a nd s t he s p in o f the e l e c t r on .

I f t he r e w e r e no s c a t t e ri ng c e n t r e s , t he e f f e c t ive H a m i l t on i a n w ou l d he

Ygen = k2 /2 m - eE . ( r + p )

( w he r e r i s t he c e n t r e o f t he e l e c t r on w a ve pa c ke t ) f o r a me t a l i n a un i f o r m

elec t r i c f i e ld E . Loca l s ca t t e r ing poten t i a l s g ive loca l t e rms in the Hami l ton ian

V r ) - A k × s ) . v v .

H e r e , t he s e c ond t e r m a r is e s f r om s p i n - o r b i t c oup l i ng i n t he l a t ti c e. A n a dd i t i ona l

con t r ibu t ion to A can a l so a ri se f rom a loca l sp in -or b i t in te ra c t ion .

There a re two d i s t inc t e f fec t s :

( a ) t he s c a t t e r i ng ma t r i x e l e me n t s be t w e e n p l a ne w a ve s t a t e s a r e e xp r e s s e d a s

(k '] V - A(k

x s ) . V V I k ) : V u , [ 1 -

iA(k x k ' ) - s ]

( by a p p l yi n g t h e g e n e r a l c o m m u t a t o r r u l e I f ( x ) , kx] ~ i 0 f ( x ) / 0 x t o

V ( r ) ) .

T hi s

me a ns t ha t t he p r oba b i l i t y o f s c a t t e r i ng

k ~ k

i s no t the same as the probabi l i ty

k ~ k

b e c a u s e o f in t e r f e r e n c e b e t w e e n t h e s p i n - o r b i t t e r m a n d t h e p o t e n t i a l

sca t t e r ing . For a weak 3 func t ion poten t i a l ,

Wkk' = V2[ 1 + 2 A Vr r n ( k x k ) . s ] ,

where n i s the dens i ty of s t a t es a t the Fermi l eve l .

Thi s skew sca t t e r ing l eads to a Ha l l cu r ren t such tha t the H a l l angle ~bn oc A V,

w h i c h i s i nde pe nde n t o f s c a t t e r i ng c e n t r e c onc e n t r a t i on , bu t w h i c h c a n be o f

e i the r s ign , depending on the s ign of V .

( b) N o w w e c o m e t o t h e s i d e j u m p t e rm .

T he t o t a l H a mi l t on i a n i s

= k 2 / 2 m + V ( r ) - e E . r + p • [ V V - e E ] ,



TRANSPORT PROPERTIES OF FERROMA GNETS 785

a n d t h e t o t a l v e l o c i t y is :

v : / ~ - i [ r , ge l =

k / m A [ V V - e E] x s p .

W i t h o u t s c a t t e r i n g , p c h a n g e s a s k i n c re a s e s u n d e r t h e i n f l u e n c e o f E , a n d

s e c o n d l y , t h e e n e r g i e s o f t h e d i f f e r e n t k s t a te s a r e a l t e r e d b y t h e s e c o n d t e r m i n v.

H o w e v e r , w h e n s c a t t e r i n g is i n t r o d u c e d , b o t h o f t h e s e c u r r e n t s a r e e x a c t ly

c a n c e l l e d o u t i n s t a t ic c o n d i t i o n s ; t h e f ir st b e c a u s e

p ) = - A t ~ ) × s - - - o ,

a n d t h e s e c o n d b e c a u s e t h e e l e c t r o n d i s t ri b u t i o n r e a d j u s t s i ts e lf t o m i n i m i z e

e n e r g y , a n d t h i s n e w d i s t r ib u t i o n a u t o m a t i c a l l y h a s a n a v e r a g e v e l o c i t y p e r -

p e n d i c u l a r t o E e q u a l t o z e r o .

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

B u t , d u r i n g e a c h s c a t t e ri n g e v e n t t h e r e is a l so a s i d e j u m p o r s h if t o f t h e c e n t r e

o f g r a v i ty o f a s c a t t e r e d w a v e p a c k e t

8 r = f 6 v

d t = - A A k × s

( a s

6 v

--- A V V × s = - A / ¢ x s d u r i n g t h e s c a t t e r i n g e v e n t ) .

N o w , t h e r e a r e t w o si d e j u m p c o n t r i b u t i o n s :

(i) e l e c t r o n s t r a v e l li n g w i th a c o m p o n e n t o f k p a r a l l e l t o E j u m p s i d e w a y s o n

b e i n g s c a t t e r e d ; t h e r e s u l t a n t o f t h e s e j u m p s i s a c u r r e n t .

(ii) e l e c t r o n s w i t h a c o m p o n e n t o f k p e r p e n d i c u l a r t o E g a i n o r lo s e a n e n e r g y

- e , 3 r . E

o n s c a t t e r i n g . T h i s s h if t s t h e t o t a l e l e c t r o n d i s t r i b u t i o n t o p r o v i d e a

s e c o n d c u r r e n t .

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

t o t a l H a l l c u r r e n t o f

2 A N e 2 E

x (s ) , w h i c h i s p r o p o r t i o n a l t o t h e e l e c t r i c fi e ld E b u t

i n d e p e n d e n t o f t h e s c a t t e r in g r a t e . T h e d e f i n it io n o f R , i s

Vy/IxMz,

w h e r e y is t h e

H a l l p r o b e d i r ec t i o n , x t h e c u r r e n t d i r e c t i o n a n d z t h e m o m e n t d i re c t io n . P u t t i n g

Vy = p l y

a n d E =

pIx,

w i t h t h e H a l l c u r r e n t j u s t g i v e n w e c l e a r l y o b t a i n R , ~ A p 2.

N o t e t h a t t h e p a r a m e t e r h r e p r e s e n t s t h e r a t e o f c h a n g e o f th e s p i n - o r b i t d i p o l e

- A k x s w i t h k . T h i s is a b a n d p r o p e r t y . H o w e v e r , l o c al s p i n - o r b i t i n t e r a c t i o n s o n

s c a t t e r in g c e n t r e s c a n g i v e a n a d d i t i o n a l c o n t r i b u t i o n t o A a n d c o m p l i c a t e t h e

p i c t u r e .

W e c a n n o w t u r n t o t h e e x p e r i m e n t a l d a ta . T h e s k e w s c at t er i n g te r m c a n b e

e x p e c t e d t o d o m i n a t e in d i lu t e a l lo y s a t l o w t e m p e r a t u r e s , a n d i n d e e d in N i b a s e d

a l l o y s f o r w h i c h p ( 0 ) ~< 1 t zf ~ cm a t h e l i u m t e m p e r a t u r e s (fig . 1 5 ) i t h a s b e e n s h o w n

t h a t t h e H a l l a n g l e ~bH i s i n d e p e n d e n t o f i m p u r i t y c o n c e n t r a t i o n , b u t d e p e n d s

s t r o n g l y o n t h e t y p e o f i m p u r i t y ( J a o u l 1 9 7 4, F e r t a n d J a o u l 1 9 72 , D o r l e i j n 1 9 76 ).

I t is p o s s i b l e t o d e f i n e H a l l a n g l e s f o r e a c h d i r e c t i o n o f s p i n , ~ b H t a n d 4~H ~ a n d

e x p e r i m e n t s o n t e r n a r y a l lo y s ( D o r l e ij n 1 97 6) o r o n th e t e m p e r a t u r e d e p e n d e n c e



786 I .A. CAMPBELL AND A. FERT

- ? H ('1 1 ~. cm)

XTRAO RDINAR Y HAL L RESISTIVITY AT /-,.2°K

10 Cu

5 Mn~Fe

Co

2

0

1 c tat*l,)

-5 Cr

lr Os

Fig. 15. Extrao rdinary Ha ll resistivity o f several types of Ni based alloys as a function of their imp urity

concentration. Th e data are limited to alloys having a resistivity smaller than abou t 1 Ixf~cm; in m ore

concentrated alloys, a side-jump contribution progressively appears and becomes predominant for

p = 10 ixf~cm (see f ig. 16) (Jaoul 1974).

o f t h e H a l l a n g l e ( J a o u l 1 9 74 ) a ll o w o n e t o e s t i m a t e t h e s e t w o H a l l a n g l e s f o r e a c h

i m p u r i t y . R e s u l t s a r e g i v e n i n t a b l e 6. T h e v a l u e s o f t h e s k e w s c a t t e r i n g H a l l

a n g l e s c a n b e d i s c u s s e d i n t e r m s o f t h e e l e c t r o n i c s t r u c t u r e o f t h e v a r i o u s

i m p u r i t i e s ( F e r t a n d J a o u l 1 9 7 2 , J a o u l 1 9 7 4 ) .

F o r s a m p l e s w i t h h i g h e r r e s i s t i v it i e s ( e i t h e r b e c a u s e o f h i g h e r i m p u r i t y c o n c e n -

t r a t i o n o r b e c a u s e t h e y a r e m e a s u r e d a t h i g h e r t e m p e r a t u r e s ) t h e s id e j u m p t e r m

b e c o m e s i m p o r t a n t . C o n s i d e r i n g o n l y d a t a t a k e n a t l o w t e m p e r a t u r e s , r e s u l ts f o r a

g i v e n a l l o y s e r ie s c a n g e n e r a l l y b e f i t t e d ( J a o u l 1 97 4, D o r l e i j n 1 9 76 ) b y t h e

TABLE 6

Skew scatte ring Hall effect in dil ute Ni ba sed alloys. Fo r each impu rity, qSH is the d ilut e limit Hal l

angle in millirad, an d ~bHt, ~bH~ are the c orres pon ding spin 1 and spin $ Hal l angles. Referen ces:

* Do rlei jn 1976, *Jaoul 1974.

Impurity Ti V Cr Mn

42H +1.5 , -4 .5 -3 , -2.5* +2.8 , +2 -6. 5 , -9 .5 t

~bH1' -3. 4* -4 *, -79 - 3*, - f - 10*

qSH~ +5.5* +6 , -3 * +4 , 3 +1.5 t

Impurity Fe Co Cu Ru Rh

~bH --6.2, --10 --6.2*, --10 .5' --10 , --23 +2.5*, +3 t 0 , --4

~bat -7 , -10 t -6 , -10 t -14 , -24* -4.7 , +3 t -1.4 , -3 t

~bH; +6 , +10 t +2.5 , +7 +3.5 , +1 0' +3 , +3 t +1.3 , -5*



TRANSPO RT PROPERTIES OF FERROM AGN ETS 787

e x p r e s s i o n

R s = a p + b p :2 ,

(33)

o r a l t e r na t i ve l y ( f i g . 16 )

c /:,H = ~ b ° B p , ( 3 4 )

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

u s u a l l y a s s u m e d t h a t t h i s r e p r e s e n t s a s e p a r a t i o n i n t o t h e s k e w s c a t t e r i n g t e r m

a n d t h e s i d e j u m p t e r m . F o r m o s t N i b a s e d a l lo y se r ie s , as w e h a v e s e e n t h e

va l ues o f 4~° v a r y c o n s i d e r a b l y , b u t t h e v a lu e s o f B h a r d l y v a r y f r o m o n e i m p u r i t y

t o a n o t h e r , w i t h B -~ - i m i l l i r a d / ~ c m . H o w e v e r , f o r t h o s e N i b a s e d a l l o y s w i t h

p + O ) / p t (0)> > 1, t h e d a t a a s a fu n c t i o n o f c o n c e n t r a t i o n c a n n o t b e r e p r e s e n t e d b y

e q . ( 3 3 ) u n l e s s o n l y a v e r y r e s t r i c t e d r a n g e o f c o n c e n t r a t i o n i s c o n s i d e r e d . I t is

i n t e r e s t i n g t o n o t e t h a t t h e s e p a r t i c u l a r a l l o y s a r e t h o s e w h i c h a l s o s h o w

a n o m a l o u s R 0 a n d r e s is ti v it y a n i s o t r o p y b e h a v i o u r a s a f u n c t io n o f c o n c e n t r a t i o n .

A t r o o m t e m p e r a t u r e , p i n N i a n d N i al lo y s is a l w a y s h i g h s o t h a t th e si d e

j u m p m e c h a n i s m c a n b e a s su m e d t o d o m i n a t e . T h e e x p e r i m e n t a l v a l u e o f t h e

r a t i o

R d p 2

i n c r e a s e s f r o m t h e p u r e N i v a l u e ,

R s / p 2 ~

0 .1 ( ~ c m G ) -1 , as a f u n c t i o n

o f im p u r i t y c o n c e n t r a t i o n a n d r a p i d ly s a t u r a t e s a t a p l a t e a u v a l u e o f a b o u t

0 .1 5 ( f~ c m G ) ~ f o r a w i d e r a n g e o f N i a l lo y s ( K 6 s t e r a n d G m 6 h l i n g 1 96 1, K 6 s t e r

a n d R o m e r 1 9 64 ), (fig . 1 7). T h e r o o m t e m p e r a t u r e

R s / p 2

v a l u e s f o r t h e a l l o y s a r e

c l o s e t o t h e v a l u e s a t lo w t e m p e r a t u r e s f o r t h e s a m e a l l o y s ( D o r l e i j n 1 9 7 6 ).

H o w e v e r , f o r c e r t a i n a l l o y s y s t e m s

R s / p 2

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

s t e a d il y w i th i m p u r i t y c o n c e n t r a t i o n . T h u s f o r N i F e , R ~ c h a n g e s s ig n a t a b o u t

1 5 % F e ( S m i t 1 95 5 , K o n d o r s k i i 1 9 6 4 ). A l l o y s w i t h t h is c o n c e n t r a t i o n o f F e s h o w

l o w v a l u e s o f R ~ / p : e v e n i f a s e c o n d h i g h r e s i s ti v i ty i m p u r i t y is i n t r o d u c e d ( L e v i n e

1961) .

I n p u r e F e a n d F e S i a l l o y s ,

R s / p 2

is r e m a r k a b l y c o n s t a n t o v e r a w i d e r a n g e o f

c o n c e n t r a t i o n s a n d t e m p e r a t u r e s ( K o o i 1 9 54 , O k a m o t o e t al. 1 96 2, w h e r e t h is

r a t i o r e m a i n s c o n s t a n t a l t h o u g h R ~ v a r i e s o v e r t h r e e d e c a d e s ) ( fig . 18 ). F o r o t h e r

F e b a s e d a l lo y s t h e r a t i o g e n e r a l l y a p p r o a c h e s t h e p u r e F e v a l u e a t m o d e r a t e o r

r

2 0

1 0 . /

1 '0 2~0

. ~ ( p ~ c m )

L °

-r 1C

t

¢ 0 2 0

9 ± (# .O _ c m )

Fig. 16. T he extraordinary Hall. angle at 4.2 K as a function of the residual resistivity of Fe A1 and

NiR u alloys (Do rleijn 1976).




-0.15

-0.1

-0.05

Rs )2 (~.cm9)._i

o ~

i

~

0

/t

o F

Ru

M o

o b

• Ti

x V

Concent ra t ion ,

I I I I I •

0 1 2 3 4 5

6

Fig. 17. The rat io

Rs p 2

in Ni and Ni a l loys a t room tem per a tu re a f te r K6 ste r e t a l. 1961 and 1964).

09

g

b.

E

o

fc~0

E

o

?

1611

156

e

A 2.04 Si-Fe /

* 3.83 Si-Fe

xO,

I I

lO 5

lo ~

Resistiv'lty ~o (D.cm)

Fig. 18 . Log -log plo t of R5 aga ins t p for Fe and som e Fe a lloys abov e ni t rogen temp era t ure a f te r

Okamoto et a l . 1962).




50

20

~ 10

o.~

o

s

0 . 5

I I I [ A

I

A12.7 % Cr IN Fe /

/

o 5.1% CF IN Fe /~/~

/ /

• 0.75 °/o CFIN Fe / /

t

x 2 .3 CF IN Fe # j / , y '

t

2 5 10 20

~ 1 0 - 8 O H M )

50

F ig . 19 . Log - log p lo t o f R s a ga in s t p f o r F e C r a l l oys , w i th t e m pe r a tu r e a s a n im p l i c i t va r i a b l e ( a f t e r

C a r t e r a nd P ugh 1966 ) .

high temperatures Softer et al. 1965, Carter and Pugh 1966). However, at low

temperatures where skew scattering can be important, the behaviour can be

completely different fig. 19) Carter and Pugh 1966). It seems that in the F__~eCr

case, there is a strong skew scattering effect at low temperatures which has

disappeared by room temperature but see Majumdar and Berger 1973). Dorleijn

1976) has made an analysis in terms of skew scattering, side jump and ordinary

Hall effect in Fe alloys at helium temperatures, but the interpretation is tricky,

particularly because samples frequently show a field dependent Hall coefficient.

The extraordinary Hall coefficient has been measured as a function of tem-

perature in pure Co Cheremushkina and Vasileva 1966).

Kondorskii 1969) suggested that the sign of the side jump effect was rela ted to

the charge and polarization of the dominant carriers, which can be compared with

the model outlined above. No satisfactory quantitative estimates of the size of the

effect seem to have been made for ferromagnetic metals, and other basic questions

concerning this mechanism remain open.

The anisotropy of the Hall effect in single crystals is technically difficult to

study, and, as a result, the existence of an anisotropy in the extraordinary Hall

coefficient of cubic metals has been uncertain. Now evidence has been provided

for the anisotropy in Rs for Fe Hirsch and Weissmann 1973) and for Ni Hiraoka

1968) at room temperature. In hexagonal Co both R0 and Rs are highly anisotro-



7 9 0 I .A . C A M P B E L L A N D A . F E R T

p i c V o l k e n s h t e i n e t a l. 19 61 ) w h i c h m e a n s t h a t m e a s u r e m e n t s o n h c p C o

p o l y c r y s t a l s a r e s u b j e c t t o s e v e r e t e x t u r e p r o b l e m s .

3.5. Thermoelectric power

I n n o n - m a g n e t i c m e t a l s u n d e r e l a s t i c s c a t t e r i n g c o n d i t i o n s , t h e t h e r m o e l e c t r i c

p o w e r T E P ) c o e f f ic i e n t d e p e n d s o n t h e d i f f e r e n t ia l o f t h e r e s is t iv i t y a t t h e F e r m i

s u r f a c e t h r o u g h t h e M o t t f o r m u l a :

d p

s = l el p

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

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

T h e T E P c u r v e s a s a f u n c t i o n o f t e m p e r a t u r e s f o r F e , C o a n d N i m e t a l s s h o w

e f f ec t s w h i c h a r e c l e a r l y d u e t o f e r r o m a g n e t i c o r d e r i n g fig . 2 0). F o r C o a n d N i ,

t h e c u r v e o f S T ) s h o w s a b u l g e to w a r d s n e g a t i v e v a l u e s o f S in t h e f e r r o m a g n e t i c

t e m p e r a t u r e r a n g e , a n d a d i s t in c t c h a r g e o f s l o p e a t T o. F o r F e , t h e b e h a v i o u r i s

s i m i l ar b u t c o m p l i c a t e d b y a p o s i t iv e h u m p i n

S T )

j u s t b e l o w r o o m t e m p e r a t u r e .

T h e c r i t i c a l b e h a v i o u r o f S T ) h a s a t t r a c t e d c o n s i d e r a b l e a t t e n t i o n . I n N i , t h e

c u r v e f o r d S / d T n e a r T c r e s e m b l e s t h e s p e c if ic h e a t c u r v e i n t h e s a m e w a y a s d o e s

d p / d T

T a n g e t a l . 1 9 7 1 ) . A l t h o u g h i t h a s b e e n a r g u e d t h a t t h e T E P a n o m a l y

r e p r e s e n t s s t r i c t l y t h e s p e c i f ic h e a t o f t h e i t i n e r a n t e l e c t r o n s T a n g e t a l . 1 97 2) a

m o r e r e a s o n a b l e i n t e r p r e t a t i o n is i n t e r m s o f t h e c r it ic a l b e h a v i o u r o f t h e e l a st ic

s c a t t e ri n g T h o m a s e t a l. 19 72 ). C o m b i n i n g t h e M o t t f o r m u l a a n d t h e e x p r e s s i o n

2 0

10

_10 ̧

T

x ¢

> - 2 0

13

-31

-41

- 5 0

i

Tc

4 0 0 T c N i ) 8 0 0 1 2 0 0

T ( K )

F i g. 20 . T h e a b s o l u t e t h e r m o e l e c t r i c p o w e r o f N i, F e , P d a n d C o L a u b i t z e t a l. 1 97 6) .




f o r t h e r e s i s t i v i t y a s a f u n c t i o n o f k n e a r T c l e a d s t o

S

=

S p - 1 A o T 1 +

Pn/P ,

w h e r e A Q =

27r2k~/3[elEv,

a n d S p is t h e b a c k g r o u n d n o n - m a g n e t i c T E P . R e s u l ts

o n G d N i 2 w e r e d i s c u s s e d in te r m s o f t h is a p p r o a c h Z o r i c e t a l. 1 9 73 ).

T h e s y s t e m a t i c s o f

S T )

w e r e s tu d i e d a t r o o m t e m p e r a t u r e a n d a b o v e in a

n u m b e r o f N i b a s e d a l lo y s V e d e r n i k o v a n d K o l m e t s 1 96 1, K o l m e t s a n d V e d e r -

n i k o v 1 9 6 2 , K 6 s t e r a n d G m 6 h l i n g 1 9 6 1 , K 6 s t e r a n d R o m e r 1 9 6 4 ) . S a t r o o m

t e m p e r a t u r e b e c o m e s r a p i d ly m o r e p o s it iv e w i th i m p u r i ty c o n c e n t r a t i o n f o r t h o se

a l l o y s f o r w h i c h

p; O)/p~ O)~<

1 ~ V , N i C r . . . ) w h il e S b e c o m e s m o r e ne g a ti ve

f o r a l l o y s w i t h

p+ O)/pt

0) ~> 1 f ig . 21 ). Th e ne ga t i v e b u l g e i n

S T )

r e m a i n s v e r y

s t r o n g f o r a w i d e r a n g e o f N i F e a l l o y s m e a s u r e d u p t o T c B a s a r g in a n d Z a k h a r o v

1 9 74 ), b u t t e n d s t o d i s a p p e a r in N i V a l l o y s V e d e m i k o v a n d K o l m e t s 1 9 61 ).

T h e l o w t e m p e r a t u r e T E P o f N i b a s e d a ll o y s h a s b e e n a n a l y z e d us in g t h e t w o

c u r r e n t m o d e l F a r r e l l a n d G r e i g 1 9 6 9, 19 7 0, C a d e v i l l e a n d R o u s s e l 1 9 71 ). I f t h e

i n t ri n s ic T E P c o e f f ic i e n t s f o r t h e t w o s p i n d i r e c t i o n s a r e S t a n d S + t h e n t h e

o b s e r v e d v a l u e o f S s h o u l d b e S = p ; S t + P t S ; ) / p ~ + p + ) a t l o w t e m p e r a t u r e s ;

a t hi g h t e m p e r a t u r e s w h e r e t h e t w o c u r r e n t s a r e m i x e d , t h e i m p u r i t y d i ff u si o n

t h e r m o p o w e r b e c o m e s S = ½ S + S + ). U s i n g t h e s e t w o e x p r e s s i o n s , F a r r e ll a n d

G r e i g 1 9 69 ) e x t r a c t e d S t , S , f o r a n u m b e r o f i m p u r i t ie s in N i a n d s im i l ar

a n a l y s e s h a v e b e e n d o n e in N i a n d C o b a s e d a l l o y s C a d e v i l l e e t a l. 1 9 68 ,

C a d e v i l l e 1 9 7 0 , C a d e v i l l e a n d R o u s s e l 1 9 7 1 ) . A d e t a i l e d d i s c u s s i o n h a s b e e n g i v e n

26

24

22

20

18

16

14

-Q 12

8

6

4

8 . 8 C r

(a )

11.

5.2

8

Ni Cr

r i i i i i . i . i

40 80 120 160 200 2z,0 280

J K )

-2

-4

-6

::k

~ ' 4 C

-1;

- - l d

2O

i

T ( K )

40 60 80 100

F i g . 2 1. T h e a b s o l u t e t h e r m o e l e c t r i c p o w e r o f s o m e n i c k e l b a s e d a l l o y s a s a fu n c t i o n o f t e m p e r a t u r e

a f t e r B e i l i n e t a l . 1 97 4 a n d F a r r e l l a n d G r e i g 1 97 0 ). a ) N i C r ; b ) N i a l l o y s .




of t he r e l a t i ons h i p be t w e e n t he e l e c t r on i c s t r uc t u r e o f t he i m pur i t y a nd t he T E P

coef f i c i en ts (Cadevi l l e an d Rou sse l 1971).

A no t he r a s pe c t o f t he t w o c u r r e n t s i t ua t i on i s t he i n f l ue nc e o f m a gnon- e l e c t r on

sca t t e r ing (Ko renb l i t and La za ren ko 1971). Sca t t e r ing of a sp in e l ec t r on to a

sp in I' s t a t e involves the c rea t io n of a ma gno n, which need s pos i t ive energ y ,

whi le sp in 1' to sp in + sca t t e r ing i s th ro ugh th e des t ruc t io n of a ma gno n. Th e

e l e c t r on - m a gnon s c a t t e r i ng w i l l t he n l e a d t o a pos i t i ve t e r m i n S a t m ode r a t e

t e m pe r a t u r e s i n a l loys w he r e t he s p in c u r r e n t dom i na t e s , a nd a ne ga t i ve t e r m

i n al l oys w he r e t he s p in 1' c u r r e n t dom i na t e s . T he T E P due t o t h i s e f f e c t w ill be

s upe r i m pos e d on t he e l a s t i c e l e c t r on - i m pur i t y t e r m e xc e p t a t ve r y l ow t e m -

pera tures , and wi l l compl ica te the ana lys i s o f the d i f fus ion t e rms . Resul t s on Ni

a l loys have been in te rpre ted wi th th i s mechani sm (Be i l in e t a l . 1974) .

A m a gno n d r a g e f f e ct ha s be e n s ugge s t e d ( B a i lyn 1962 , G ur e v i c h a nd K or e n b l i t

1964 , B la t t e t a l . 1967) . Measurements on the TEP in a NiCu and a NiFe a l loy in

a pp l i e d f i e l d s a ppe a r t o be c ons i s t e n t w i t h t h i s m e c ha n i s m ( G r a nne m a n a nd

B e r ge r 1976 ) . H ow e ve r , t he s t r ong pos i t i ve T E P hum p i n pu r e F e doe s no t ha ve

this origin (Blatt 1972).

T he va l ue o f S i s a n i s o t r op i c w i t h r e s pe c t t o t he m a gne t i z a t i on d i r e c t i on i n a

f e r r o m a g n e t . M e a s u r e m e n t s o n F e a n d N i s i n g l e c r y s t a l s a t r o o m t e m p e r a t u r e

( M i ya t a a nd F una t oga w a 1954) ga ve

AS100 = + 0.70 Ix V/ K, AS m -- - 0.13 txV/K in F e ,

a n d

AS~00 = +0.5 7 ~xV/K, AS m = +0 .69 txV/K in N i .

The Fe resu l t was conf i rmed by B la t t (1972) .

3 . 6 . N e r n s t - E t t i n g s h a u s e n e f f e c t

T hi s i s t he t he r m oe l e c t r i c a na l ogue o f t he H a l l e f f e c t . I t ha s be e n s t ud i e d i n t he

pu r e f e r r om a gne t i c m e t a l s a nd i n a num be r o f a l l oys ( I va nova 1959 , K ondor s k i i

and Vas i l eva 1964, Cheremushkina and Vas i l eva 1966, Kondorsk i i e t a l . 1972 ,

Vas i l eva and Ka dy rov 1975). L ike Rs , th i s coe f f i c i en t va r i es s t rongly wi th t em-

p e r a t u r e i n f e r r o m a g n e t s . K o n d o r s k i i ( 1 9 6 4 ) p r o p o s e d t h e p h e n o m e n o l o g i c a l

r e l a t i ons h i p

Q = - a + jgp )T ,

a nd t he o r i g in o f t he e f f e c t w a s d i s cus s e d in t e r m s o f t he s i de j um p m e c h a n i s m by

Berger (1972) and Campbe l l (1979) .

3 . 7. Th erm al co nduc t i v i t y

This i s no t a pure ly e l ec t ron t ranspor t e f fec t , a s hea t can be ca r r i ed a l so by

phonons a nd e ve n m a gnons , a nd s e pa r a t i ng ou t t he d i f f e r e n t c on t r i bu t i ons i s

d i f f i cu l t . Fa r re l l and Gre ig (1969) in ca re fu l measurements on Ni and Ni a l loys

ha ve s how n t ha t a c ohe r e n t a na l y s i s o f t he a l l oy da t a ne e ds t o t a ke i n t o a c c oun t




the two c u r r e n t c ha r a c te r o f t he c onduc t ion . The y f ound tha t i t wa s no t poss ib l e

to de c ide f o r o r a ga ins t t he p r e se nc e o f a ny e l e c t r on - e l e c t r on t e r m in pu r e N i a t

low tempera tures (Whi te and Ta insh 1967) .

At h igher tempera tures , Tursky and Koch (1970) have shown tha t i t i s poss ib le

to use the spon ta ne ous r e s i s t iv i ty a n i so t r opy to se pa r a t e ou t phonon a nd e l e c t r on

the r m a l c onduc t iv i ty .

By mea sur em ent s in s t rong fie lds , Ye lo n a nd Be rger (1972) ident i f ied a magn on

contr ibu t ion to the low tem per a tu re the rm al con duc t iv i ty in N_iiFe.

The the r m a l c onduc t iv i ty o f N i shows a n a b r up t c ha nge o f s lope a t Tc ( La ub i t z

et a l . 1976). Th is pro pe r ty is very dif f icult to m eas ure with hig h precision.

4 Dilute ferromagnetic alloys

4.1 . Pal ladium based a l loys

I t ha s b een kn ow n for som e t ime tha t P dF e, Pd Co , Pd M n an d P__ddNi a l loys are

g ia n t m om e n t f e r r om a gn e t s a t l ow c onc e n t r a t ions ; t he t r a nspor t p rope r t i e s o f

the se sys t e m s ha ve b e e n we l l s tud ie d .

4.1.1. Res ist ivi ty an d isotropic m agnetore sistance

P dF e a l loys ar e so f t f e r r om a gne t s dow n to a t l e a st 0 .15% F e . T he F e m a g-

ne t iza t io n a t T ~ Tc sa tura tes com ple te ly in small appl ied f ields (Ch out eau and

Tournie r 1972, Howar th 1979) . The magne t ic d isorder a t r e la t ive ly low tem-

pera tu res i s in the form of magn ons ; for the d i lu te a l loys (C < 2% Fe) , i t appears

tha t t he m a g non - e le c t r o n sc a t te r ing is e sse n t ia l ly inc ohe r e n t so the m a gne t i c

r e s i s t iv i ty i s p r opor t iona l t o the num be r o f m a gnons p r e se n t , l e a d ing to a

t e m p e r a tu r e de p e nd e n t r e s i st ivi ty p r opor t iona l t o

T 3/2

for T ~ Tc an d a ch arac-

t e r i s t i c t e m pe r a tu r e de pe nde n t ne ga t ive m a gne to r e s i s t a nc e ( Long a nd Tur ne r

1970, Will iams and Loram 1969, Will iams et a l . 1971, Hamzi6 and Campbell

1978). At h igher con cent ra t ions a T 2 res is tiv i ty va r ia t ion replaces the

T 3 2

behaviour (Ska lski e t a l . 1970) . At the Cur ie tempera ture the re i s a change in

s lope of the

p T )

curv e bu t i t is dif f icult to a naly ze the res ults in term s o f cr it ical

sca t te r ing beha vio ur becaus e of smear ing d ue to the sp read o f Tc va lues in the

samples (Kawatra e t a l . 1969) .

P dM n a lloys a r e f e r r o m a gne t s be low 4% M n c onc e n t r a t ion in tha t t he y show

a high in i t ia l suscept ib i l i ty be low a wel l de f ined order ing tempera ture (Raul t and

Burg er 1969, Co les e t a l . 1975). In fac t , h igh fie ld mag ne t iza t ion m easu rem ents

(S ta r e t a l . 1975) show tha t the Mn magne t iza t ion only becomes t rue ly sa tura ted

whe n ve r y s tr ong m a gne t i c fi elds a r e a pp l ie d . The t e m p e r a tu r e de pe n de nc e o f the

res is t iv i ty of these a l loys i s qua l i ta t ive ly s imi la r to tha t observed in PdFe , wi th a

change of s lope in

p T )

at Tc and a

T 3/2

var ia t ion of the res is t iv i ty a t low

tem pera ture s (Wil l iams and Lor am 1969). In cont ras t to the Pd Fe a l loys the

m a gne to r e s i s t a nc e r e m a ins s t r ong ly ne ga t ive e ve n whe n T t e nds to z e r o ( Wi l li a m s

et al. 1973).



794 I .A. CAMP BELL AND A. FERT

P d C o a l lo y s h a v e v e r y s i m il ar o r d e r in g t e m p e r a t u r e s a n d t o t a l m a g n e t i c

m o m e n t s p e r a t o m a s t h e P d F e a l l oy s ( N i e u w e n h u y s 1 975 ), a n d t h e t e m p e r a t u r e

d e p e n d e n c e o f t h e r e s i s ti v i ty is a g a in o f t h e s a m e t y p e ( W i ll ia m s 1 97 0). H o w e v e r

t h e p a r a m a g n e t i c r e s i s ti v i ty a t T > Tc i s p r o p o r t i o n a l t o t h e C o c o n c e n t r a t i o n

(Colp and Wil l iams 1972) whereas in P__ddFea l l o y s i t i n c r e a s e s a s t h e s q u a r e o f t h e

F e c o n c e n t r a t i o n ( S k a ls k i e t a l. 19 70 ). T h e P d C o a l lo y s b e l o w 5 C o s h o w a

n e g a t i v e m a g n e t o r e s i s t a n c e a t T ~ T c w h i c h i n d i c a te s t h a t t h e y a r e n o t t r u e

f e r r o m a g n e t s ( H a m z i 6 e t a l . 1 9 7 8 a ) * .

P d N i al l o y s a r e f e r r o m a g n e t s a b o v e a c r it ic a l c o n c e n t r a t i o n o f 2 . 3 N i (T a r i

a n d C o l e s 1 97 1). N e a r t h i s c o n c e n t r a t i o n t h e l o w t e m p e r a t u r e v a r i a t i o n o f t h e

r e s i s t iv i t y o f t h e a l l o y s b e c o m e s p a r t i c u l a r l y s t r o n g ( T a r i a n d C o l e s 1 9 7 1) . B o t h

t h e p a r a m a g n e t i c a n d f e r r o m a g n e t i c a l l o y s s h o w a l a r g e p o s i t i v e m a g n e t o r e s i s -

t a n c e d u e t o a n i n c r e a s e i n t h e l o c a l m o m e n t s a t t h e N i s it es w i th t h e a p p l i e d f ie l d

(Gen icon e t a l . 1974 , Hamz i6 e t a l . 1978a ) .

4 1 2 M agn etoresistan ce anisotropy

P d F e , P__d_dCoa n d P d N i a l l o y s a ll s h o w p o s i t i v e a n i s o t r o p i e s P ll > P ± a t m o d e r a t e

m a g n e t i c i m p u r i t y c o n c e n t r a t i o n s . A t l o w c o n c e n t r a t i o n s P_ddFe s a m p l e s s h o w

v a n i s h i n g l y s m a l l a n i s o t r o p i e s ( H a m z i 6 e t a l. 1 97 8a). F r o m t h is a n d o t h e r e v i d e n c e

i t h a s b e e n c o n c l u d e d t h a t t h e C o a n d N i i m p u r i t i e s c a r r y l o c a l o r b i t a l m o m e n t s .

4 1 3 Extraordina ry H all e f fect

O v e r a b r o a d c o n c e n t r a t i o n r a n g e t h e H a l l co e f f ic i e n t i n P d F e a l lo y s b e h a v e s

s i m i la r l y t o t h a t in c o n c e n t r a t e d N i F e a l lo y s , c h a n g i n g si g n n e a r 2 0 F e ( M a t v e e v

e t a l. 1 97 7, D r e e s e n a n d P u g h 1 96 0). A t l o w c o n c e n t r a t i o n s th e H a l l a n g l e te n d s

to zero for PdFe and P__d_dMnb u t t a k e s o n a c o n c e n t r a t i o n i n d e p e n d e n t v a l u e f o r

P d N i a n d P d C o ( H a m z i 6 e t a l. 1 9 78 b, A b r a m o v a e t a l . 1 9 74 ). T h i s s h o u l d b e

r e l a t e d t o t h e l o c a l o r b i t a l m o m e n t s o f C o a n d N i i m p u r i t i e s .

4 1 4 Thermoelectric po we r

I n t h e c o n c e n t r a t e d f e r r o m a g n e t s , f e a t u r e s c l e a r l y a s s o c i a t e d w i t h t h e f e r r o -

m a g n e t i c o r d e r in g a r e v i s ib le in t h e t e m p e r a t u r e d e p e n d e n c e o f t h e T E P . F o r t h e

P d b a s e d a l lo y s t h is d o e s n o t s e e m t o b e t h e c a s e e x c e p t p e r h a p s w h e n t h e

m a g n e t i c i m p u r i t y c o n c e n t r a t i o n is g r e a t e r t h a n 5 ( G a i n o n a n d S i e r ro 1 97 0). A t

1 , o r lo w e r , c o n c e n t r a ti o n s P d F e a n d P d M n s h o w w e a k n e g a t iv e o r p o s it iv e

T E P b e l o w 2 0 K v a r y i n g i n a r a t h e r c o'--m p le x w a y w i t h c o n c e n t r a t i o n a n d

t e m p e r a t u r e ( G a i n o n a n d S i e r r o 1 9 7 0 , M a c d o n a l d e t a l . 1 9 6 2 , S c h r o e d e r a n d U h e r

1 97 8) . P_dd 1 C o s h o w s a n e g a t i v e T E P h u m p a t 2 0 K ( G a i n o n a n d S i e r r o 1 9 70 );

t h i s h u m p b e c o m e s m o r e p r o n o u n c e d a n d g o e s t o l o w e r t e m p e r a t u r e s a s t h e

c o n c e n t r a t i o n i s d e c r e a s e d ( H a m z i 6 , 1 9 80 ). B e l o w t h e c r it ic a l c o n c e n t r a t i o n P d N i

a l lo y s s h o w a s t r o n g n e g a t i v e h u m p i n th e T E P a r o u n d 1 5 K w h i c h d i s a p p e a r s o n c e

t h e c o n c e n t r a t i o n e x c e e d s t h e c r i t i c a l v a l u e ( F o i l e s 1 9 7 8 ) .

* They can he considered to be quasiferr omagnets , i.e. , systems having an overall magnetic moment

but where the local moments are each somewhat disoriented with respect to the average moment

direction.



TRANSPORT PROPERTIES OF FERROM AGNETS 795

4.2. Platinum based alloys

A g ain , Pt__Fe a n d Pt___Coa r e g i a n t m om e n t f e r r om a gne t s a t c onc e n t r a t ions o f a f e w

percent , but a t lower concent ra t ions the behaviour i s more compl ica ted . For P t__Fe

below abou t 0.8 spin glass ord er se ts in (O do do 1979).

In the fe r romagne t ic concent ra t ion range the re i s the usua l s tep in p T) a t the

order in g tem pera ture , but be low 0 .8 Fe th is s tep d isappears (Lo ram e t a l. 1972).

The i so t ropic magne tores is tance i s s t rongly nega t ive a t concent ra t ions less than

ab ou t 5 Fe (Ham zi6 e t a l . 1981).

P tC o a l loys be low 1 Co show res ist iv i ty va r ia t ions which a re com plex because

o f c om pe t ing t e nde nc ie s to Kondo c onde nsa t ion a nd to m a gne t i c o r de r ing ( R a o e t

a l . 1975, W ill iams et a l . 1975) . A t conc ent ra t i on s abo ve abou t 1 Co a step can

be seen in p T) a t To. The i so t ropic magne tores is tance i s pos i t ive a t low

con cen tra t io ns, becom ing negati ve by 2 Co (Lee e t a l . 1978, Ham zi6 e t a l. 1980).

Both Pt__Fe and P tC o a l lo ys show con cent ra t ion ind epe nd ent res ist iv i ty an iso-

t ropies and ext raordinary Hal l angles a t low concent ra t ions (Hamzi6 e t a l . 1979) .

The low t e m pe r a tu r e the r m oe le c t r i c powe r o f P tC o a l loys be c om e s s t r ong ly

nega t ive be low abou t 2 Co concen t ra t io n (Lee e t a l. 1978). This TE P is sens i t ive

to appl ied magne t ic f ie lds .

Pt M n alloy s are spin g lasses (Sarkiss ian an d T ay lo r 1974), an d Pt___Ni alloy s are

no t m a gne t i c a l ly o r de r e d be low 42 Ni .

5 Amorp hous alloys

Since the ea r ly 1970s cons iderable e f for t has been devoted to the s tudy of the

e le c t r i c a l a nd m a gne t i c p r ope r t i e s o f a m or phous a l loys . The r e s i t i v i ty m in im um

obse r ve d in m a ny sys t e m s ha s be e n sub je c t t o m uc h c on t r ove r sy .

5.1. Resistivity of amorphous alloys

Th e am orp hou s a l loys have a ve ry h igh res is t iv i ty (p ~ 100 Ix l lcm) which ch anges

re la t ive ly l i t tle a s a func t ion of te mp era tu re . F igure 22 shows tha t , in se r ies of NiP

a l loys , the tempera ture coef f ic ien t changes f rom pos i t ive to nega t ive as the

c onc e n t r a t ion o f P incr e ase s . Th i s be ha v iou r i s we l l e xp la ine d in the Z im a n m od e l

of the res is t iv i ty of l iquid meta ls (Ziman 1961) and i t s ex tens ion to amorphous

a l loys (Nage l 1977). In th e Z im an mo de l the res is t iv i ty turns o ut to b e propor -

t iona l to a 2kv) w h e r e kv i s t he F e r m i wa ve ve c to r a nd a q) t he a tom ic s t r uc tu r e

factor . I f 2kv is c lose to the f irs t peak of a q), the res ist iv i ty is h igh and decreases

as a func t ion of T owing to the the rmal broadening of the p~ak. In cont ras t , i f

2kv

l ie s we l l be low (or we l l above) the peak , the res is t iv i ty i s r e la t ive ly low and

increases as a function of T. In the NiP al loys ( f ig. 22) the addit ional conduction

e lec t rons provided by the h igher concent ra t ions of P ra ise 2kF to the f i r s t peak of

a q), which accounts for the exper imenta l behaviour (Cote 1976) . On the o ther

han d, th e smal l r es ist iv i ty upt urns observ ed in NiP a t low temp era ture (fig . 22)




n

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T E MP E R A T U R E ( K )

F ig . 22 . R e l a t i ve r e s i s t i v i t i e s o f a m or phous a nd c r y s t a l l i ne N i - P a l l oys . N om ina l c om pos i t i ons a r e

i n d i ca t ed ; P am / p c rys t )2 9 3 ~ 3 . T he i n se t show s p a nd dp dT a s a f unc t ion o f t he c om pos i t i on a f t e r C o te

1976) . S im i l a r r e su l t s a re g ive n by B ouc h e r 1973 ) f o r N i - P d - P a m o r ph ou s a l loys .

cannot be explained by the Ziman model. Such resistivity upturns, which

generally give rise to a resistivity minimum, have been found in many amorphous

systems. They have been found in both ferromagnetic and non-ferromagnetic

amorphous alloys and, up to now, only in alloys containing transition or rare-

earth) metals. Their origin has been subject to much controversy.

Resistivity minima have been first found by Hasegawa and Tsui 1971a, b) in

amorphous PdSi containing Cr, Mn, Fe or Co impurities fig. 23). The classical

features of the Kondo effect are observed: the resistivity varies logarithmically over

a large temperature range and becomes constant in the low temperature limit; at

low concentration of magnetic impurities the logarithmic term increases with the

concentration, there is a negative magnetoresistance. But, surprisingly, the resis-

tivity minimum still exists in the most concentrated alloys which are ferro-

magnetic. These results seem to indicate that weakly coupled moments subsist in

amorphous ferromagnets and can give rise to Kondo scattering. Results on many

other systems have suggested that the coexistence of ferromagnetism and Kondo

effect is quite general in amorphous alloys; thus large logarithmic upturns have

been observed fig. 24) in ferromagnets of the series FeN•B, FeNiPB, FeN•PC,

FeNPBS Cochrane et al. 1978, Babi6 et al. 1978, Steward and Phillips 1978),

FeNiPBA1, FeMnPBA1, CoPBA1 Rao et al. 1979), PdCoP Marzwell 1977); in




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and Tsuei 1970b).

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Fig. 24 . Res is t iv i ty ve rsus log T plo t for two fe rr oma gne t ic a mo rph ous a l loys : A) Fe75P16B6A]3 and

O) Fe60NilsP16B6AI3 after Rapp et al. 1978).




m a n y c a s e s t h e a d d i t i o n o f s m a l l a m o u n t s o f C r s t r o n g l y e n h a n c e s t h e r e s i s t i v i t y

u p t u r n .

O n t h e o t h e r h a n d , C o c h r a n e e t a l. 1 97 5) f o u n d t h a t th e l o g a r i t h m i c r e s is t iv i ty

u p t u r n o f se v e r a l a m o r p h o u s a l l o y s w a s f ie l d i n d e p e n d e n t , i n c o n t r a s t t o w h a t i s

g e n e r a l l y o b s e r v e d i n K o n d o s y s t e m s . T h e y a l s o n o t i c e d a l o g a r i t h m i c u p t u r n i n

N i P a l l o y s w i t h h i g h P c o n c e n t r a t i o n i n w h i c h t h e N i a t o m s w e r e n o t s u p p o s e d t o

c a r r y a m a g n e t i c m o m e n t . O n t h e b a s i s o f t h e s e o b s e r v a t i o n s t h e y r u l e d o u t t h e

e x p l a n a t i o n b y t h e K o n d o e f f e c t a n d p r o p o s e d a n o n - m a g n e t i c m e c h a n i s m . T h e i r

m o d e l t r e a t s t h e e l e c t r o n s c a t t e r i n g b y t h e t w o l e v e l s y s t e m s w h i c h a r e s u p p o s e d

t o b e a s s o c i a t e d w i t h s t r u c t u r a l i n s t a b i l i t i e s i n a m o r p h o u s s y s t e m s ; a v a r i a t i o n o f

t h e r e s i s ti v i ty i n - l n T 2 + A 2) i s p r e d i c t e d , w h e r e A i s a m e a n v a l u e o f t h e e n e r g y

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

a l l o y s f i t r a t h e r w e l l w i t h s u c h a v a r i a t i o n l a w .

A t t h e p r e s e n t t i m e 1 97 9) h o w e v e r t h e t r e n d i s i n f a v o u r o f a n e x p l a n a t i o n o f

t h e r e s i s t i v i t y m i n i m a b y t h e K o n d o e f f e c t r a t h e r t h a n b y a n o n - m a g n e t i c

m e c h a n i s m . C l e a r e x a m p l e s o f l o g a r i t h m i c r es i st i v it y u p t u r n s i n n o n - m a g n e t i c

s y s t e m s a r e s t il l l a c k in g : a l l o y s s u c h a s N i P o r Y N i c a n b e s u s p e c t e d t o c o n t a i n

m a g n e t i c N i c lu s t e r s B e r r a d a e t a l. 19 78 ). O n t h e o t h e r h a n d , s y s t e m a t i c s t u d i e s

o f t h e re s i st iv i t y o f F e N i P B B a N d e t al . 1 97 8), F e N i P B A I , F e M n P B A 1 R a o e t a l.

1 97 9) h a v e s h o w n d e f i n i t e c o r r e l a t i o n s b e t w e e n t h e r e s i s ti v i ty a n o m a l i e s a n d t h e

m a g n e t i c p r o p e r t i e s l o g a r i t h m i c t e r m l a r g e w h e n T c i s s m a l l , e t c .) ; it h a s b e e n a l s o

f o u n d i n s e v e r a l s y s t e m s t h a t t h e l o g a r i t h m i c u p t u r n i s l o w e r e d b y a n a p p l i e d

f ie ld . F i n a l ly , M 6 s s b a u e r e x p e r i m e n t s o n F e N i C r P B a l lo y s h a v e f o u n d v e r y s m a l l

h y p e r f i n e f i e l d s o n a s i g n i f i c a n t n u m b e r o f F e s i t e s , w h i c h s e e m s t o c o n f i r m t h e

c o e x i s t e n c e o f f e r r o m a g n e t i s m a n d K o n d o e f f e c t C h i e n 1 97 9).

W h a t w e h a v e w r i tt e n u p t o n o w c o n c e r n e d t h e m e t a l - m e t a l l o i d a ll o y s w h i c h

h a v e b e e n t h e m o s t s tu d i e d a m o r p h o u s a ll o ys . S t u d ie s o f m e t a l - m e t a l a m o r p h o u s

a l lo y s o f r a r e - e a r t h s w i t h t r a n s i ti o n o r n o b l e m e t a l s h a v e b e e n a ls o d e v e l o p e d

r e c e n t l y . R e s i s t iv i t y m i n i m a h a v e b e e n a g a i n o b s e r v e d i n t h e s e s y s t e m s b u t a p p e a r

t o b e g e n e r a l l y d u e t o c o n t r i b u t i o n s f r o m m a g n e t i c o r d e r i n g a n d n o t t o K o n d o

e f fec t . In N i3Dy f ig . 25 ) the re s i s t iv i ty inc rea se s e i th e r i f a mag ne t i c f i e ld i s

a p p l i e d o r if t h e t e m p e r a t u r e is l o w e r e d b e l o w t h e o r d e r i n g t e m p e r a t u r e T o. T h i s

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

c o n t r a s t t o w h a t i s o b s e r v e d i n c r y s t a l l i n e f e r r o m a g n e t i c s . T h i s h a s b e e n a s c r i b e d

b y A s o m o z a e t a l. 1 9 77 a, 1 9 78 ) t o c o h e r e n t e x c h a n g e s c a t t er i n g b y t h e r a r e - e a r t h

s p in s N i h a s n o m a g n e t i c m o m e n t i n t h e s e a l lo y s ). T h e m o d e l c a l c u l a t i o n p r e d i c t s

a r e s is t iv i t y t e r m p r o p o r t i o n a l t o m 2 k v ) w h e r e r e ( q ) i s t h e s p i n c o r r e l a t i o n

f u n c t i o n

m ( q ) = N C e l j ( J + 1 ) R , ~ , e x p [ iq • ( R - R ' ) I J R J R ' .

H e r e C1 is t h e c o n c e n t r a t i o n o f m a g n e t i c io n s , h a v i n g lo c a l m o m e n t s J a n d p l a c e d

a t R , R ' ; t h e s u m is o v e r t h e p a i r s o f m a g n e t i c i o n s.

T h e r e s i st iv i ty w i ll d e p e n d o n t h e m a g n e t i c o r d e r t h r o u g h m ( 2 k v ) ; f o r e x a m p l e ,




2 9 6

2 9 5

I I I

Dy Ni 3

o H : O k G

~ H : 8 k G

H : 2 O k G

o H : 3 O k G

t 1 1 I I I

799

u 2 9 4

>

2 9 3

u~

Q~

2 9 2

2 9 1

I X x g~ •

t

/

I I

2 0

3 ~ _

29 2 I I I

50 100 150 200 250

I I

4 0 6 0 8 0 T K )

F ig. 25 . T h e r e s i s ti v i t y o f a D yN i3 a m or pho us a l l oy i n s e ve r a l a pp l i e d f i e lds is p lo t t e d a s a f unc t ion o f

t e m pe r a tu r e a f t e r A s om oz a et a l. 1977 ).

f e r roma gne t i c c o r r e l a t i ons w i l l i nc r e a s e o r de c r e a s e p a c c o rd i ng t o w he t he r t he

i n t e r f e r e nc e s a r e c ons t ruc t i ve o r de s t ruc t i ve . T he N i 3 -R E a l l oys s hou l d c o r -

r e s pond t o t he c a s e o f f e r roma gne t i c c o r r e l a t i ons a nd c ons t ruc t i ve i n t e r f e r e nc e s .

T h e A g - R E , A u - R E a n d A I - R E a m o r p h o u s a ll oy s a ls o s h ow a c le a r c o n t ri b u ti o n

f rom ma gne t i c o rde r i ng t o t he r e s i s t i v i t y , bu t t he i n t e rp r e t a t i on s e e ms t o be a

l it tl e mo re c om pl i c a t e d t ha n fo r t he N i 3 -R E a l loys A s om oz a e t al . 1979 , F e r t a nd

A s om oz a 1979) . F i na ll y , a l loys o f t he s er i e s F e -R E a nd C o -R E ge ne ra l l y s how a

m o n o t o n i c d e c r e a s e o f th e r e s is ti v it y f r o m t h e h e l i u m r a n g e t o r o o m t e m p e r a t u r e

Coc hra ne e t a l. 1978, Ze n e t a l . 1979). In these a l loys of h igh Tc the v a r i a t ion of

t he r e s i s t i v i t y due t o ma gne t i c o rde r i ng mus t be d i s p l a ye d ove r a w i de t e m-

pe ra t u r e r a nge a nd is c e r t a i n l y di ff ic u lt t o s e pa ra t e f r om t he no rm a l va r i a t i on d ue

t o t he phonons a nd t he t he rma l va r i a t i on o f t he s t r uc t u r e f a c t o r . We be l i e ve t ha t

t h i s no rma l va r i a t i on s hou l d be p r e domi na n t , s pe c i a l l y a t l ow t e mpe ra t u r e .

S imi la r ly , in the a l loys such as FeNiPB di scussed above , a cont r ibut ion f rom

m a g n e t i c o r d e r i n g t o

p T)

c e r t a i n l y e x i s t s bu t i s l i ke l y c ove re d up by o t he r

c on t r i bu t i ons K o ndo o r st r uc t u r a l e f f ec t s) a t l ow t e m pe ra t u r e .




5.2. Hall effect and resistivity anisotropy of amorphous alloys

The amorphous ferromagnetic alloys have a very large extraordinary Hall effect

which generally covers up the ordinary Halt effect. This is because the extra-

ordinary Hall resistivity, in contrast to the ordinary one, is an increasing function

of the scattering rate the contributions from skew scattering and side-jump are

roughly proport ional to p and p2 respectively). Thus pi~ B) is practically propor-

tional to the magnetization in many systems and, for example, is frequently used

to record hysteresis loops McGuire et al. 1977a, b, Asomoza et al. 1977b).

The extraordinary Hall effect of ferromagnetic alloys of gold with nickel, cobalt

or iron has been studied by Bergmann and Marquardt 1979) and ascribed to skew

scattering; the change of sign of pn between Ni and Fe has been accounted for by

a model based on a virtual bound state picture of the 3d electrons. On the other

hand, the extraordinary Hall effect of FeNiPB alloys rather suggest a side-jump

mechanism Malmhfill et al. 1978). The extraordinary Hall effect has been also

studied in amorphous alloys of transition metals with rare-earths and related to

the magnetization of the transition and rare-earth sublattices in phenomenological

models Kobliska and Gangulee 1977, McGuire et al. 1977, Asomoza et al.

1977b).

The spontaneous resistivity anisotropy is rather large in amorphous alloys of

gold with nickel or cobalt /911- p± -~ 1 ix12cm) and has been interpreted in a model

of virtual bound state for the 3d electrons Bergmann and Marquardt 1979). The

resistivity anisotropy seems to be smaller in alloys of the FeNiP type Marohnid et

al. 1977). The resistivity anisotropy has been also studied in amorphous alloys of

nickel or silver with rare-earths and turns out to be mainly due to electron

scattering by the electric quadrupole of the 4f electrons Asomoza et al. 1979).

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transport properties of ferromagnets

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