A_FERT_Anisortropic_Scattering.pdf

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Journal of Magnetism and Magnetic Materials 5 (1977) 23 -3 4 North-ltoUand Publishing Company

SP O NTANEO US RESISTIVITY ANISO TRO P Y IN Ni ALLO YS

O . J A O U L , I .A , C A M P B E L L a n d A . F E R TLaboratoire d e Phys ique des S ol ids *, Universi tY"Paris-Sud, 9 ! 405-Orsay, FranceReceived 28 September 197 6, in revised form 4 Novem ber 1976

We present experim ental results for the resistivity anisotropy of N i alloys containing transition imp urities, rro m these re-sults and those of other authors, we show that the Smit spin m ixing mechanism is dom inant in all cases where there is no dvirtual bo un d state at the spin t Fermi lev el. When such a state is present, an additional mechan ism due t o th e L z S z lYart ofthe sp in-o rbit interaction m ust be included in the analysis.

1. Introduction

T he spon t a n e ous a n i so t r opy o f t he r e s is t iv i t y o f fe r-r oma gne t i c me ta l s ha s be e n kno w n f o r w e l l ove r a ce n-tu r y , bu t a l t hough i t has be e n ge ne r a ll y r e c ogn i z e dtha t t h i s is a sp in - o r b i t e f f e ct , va r ious t ype s o f me -c ha n ism ha ve be e n p r opose d t o e xp l a in t he r e l a t ionb e t w e e n th e s p i n - o r b i t c o u p li n g a n d t h e c o n d u c t iv i t y .Re c e n t l y , in t e r e s t i n t he e f f e ct ha s be e n r e ne w e d be -c a use o f t he p r a c t i c a l a pp l i c a ti ons ( bubb l e me m or yread-outs , t ape readers) .

We wi l l d i scuss the sp ontan eou s res i s t iv i ty aniso-t ropy in Ni ba~ed a l loys . In a fe r romagnet ic meta l ,and in par t icula r in Ni , spin t and spin ~, e lec t rons ( i .e .sp in pa r a ll e l a nd a n t i pa ra l l el t o t he ma g ne t i z a t i on)car ry cur rent in para l le l and one can def ine a t T = 0spec if ic res idua l r es i s t iv i t ies Po t and Po ~- The ra t io

= P0 ~ /p 0 t f o r a g ive n impu r i t y i n a g ive n hos t c a nbe e s t ima te d by me a sur e me nt s on t he t e mpe r a tu r e de -pendence of a l loy res i s t iv i t ies or by te rnary a l loy re -s idua l r es i s tiv i ty mea surem ents [ 1 -5 ] .In the present work we examine in de ta i l the re la -t i on be tw e e n t he r e s i st iv i ty a n i so t r opy a nd t he tw oba nd c ha r a c t e r o f t he c ond uc t i on . We ha ve c 0 rr ie d ou te xpe r ime n t s on a w ide r r a nge o f b ina r y N i ba sed a l l oystha n i n p re v ious w or k , t oge the r w i th c e r t a in t e r na rya l loy sys t ems . We ha ve pa id pa r ti c u l a r a t t e n t i on t o t hein f lue nc e o f t e x tu r e a nd o f pa r a si t ic im pur i t i e s on t heme a sur e d a n i so t r opy . We f ind t ha t f o r e a c h impu r i t y

* Laboratoire associt~ au CNR S.23

the r e i s a c ha ra c t e ri s t ic c o nc e n t r a t i on i nde p e nde n t val -ue of the anisot ropy; for low res idua l r es i s t iv i ty im-pur i t i e s t he a n i so t r opy doe s no t s a tu r a t e a t a c on-c e n t r a t i on i nde p e nde n t va lue un t i l ~the c onc e n t r a t i onis such tha t th e res idua l r es i s t iv i ty is grea te r th anabout 2/a~2cm. We revise upwards previous es t imateso f t he c ha r a c te r i st i c va lue s o f t he a n i so t r opy f o r N i C oa n d N i F e . For cer ta in N i C o samples the anisot ropieswe have measured a re the la rges t ever repor ted on anys y s t e m .We f ind tha t for impur i t ies wi th va lues of a grea te rt ha n 1 t he a n i so t r opy i s p r opor t i ona l t o ( a - I ) c on-f i r ming tha t a Smi t [ 6 ] sp in m ix ing t ype o f me c ha n i smis the dominant one in these a l loys [7] . However , ina l loys where a i s less than 1 , a l thoug h the an isot ro pyis a lways smal l , there a re devia t ions f rom the (a - l ) rule .T o e xp l a in t he se de v i a t i ons w e p r opose a n a dd i t i ona li n t r a - sp in - ba nd L z Sz sp in - o r b i t me c ha n i sm w hic hplays a role in the presence of a vi r tua l s ta te a t thesp in t Fe r mi le ve l. Re su l t s on t he a n i so t r opy o f e a c hsp in ba nd se pa r a te ly o b t a ine d f rom a n a na lys is o f t e r-na r y a l l oy da t a [ 8 ] sup por t t h i s a ppr oa c h .2. D ef init ion of t i l e spon t a ne ous ma gne to r e s i s t a nc e

I n a dd i t i on t o t he w e l l - know n o r d ina r y ma gne to -res i s tance e f fec t observed in a l l meta ls , magnet ic met -a ls have a n a dd i t i ona l t e r m de pe nd ing on t he r e la t iveor i e n t a t i on o f t he ma gne t i z a t i on a nd t he c u r r e n t . Fora f u ll y ma gne t i z e d f e r r oma gne t i c p o lyc r ys t a l t he r e-s i s t iv i ty i s of the formp( B , O) = - rio + ( cs2 0 - ~ ) A p + S o ( B , 0 ) , ( l )


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2~where 0 i s the angle between the current d i rec t ion andt i le macroscopic mag net iza t ion of the sam ple . The las tterm represents the ord inary Lorentz force magnetore-s i s tance which depends on the magnet ic induc t ion 8 ;the second term is the spontaneous magnetoresis tance~ . To sepa ra t e t he two t e rms expe r imen ta l l y , t hesample res is tance i s measured as a fun ct ion of appl ied

! Then , the curves for p0(B ) andie ld for 0 = 0 and ~ .p~(B) above technica l sa tura t ion are ex t ra pola ted backto B - 0 . e l iminat ing the Lorentz te rm , f ig . 1 . We de-f ine the spontane ous m agnegoresis tance para m eterAp/,~ byap = p,,(s o) - pi (B --, o) (2)

~PlI(B -* O) + }p (B -+ O)in a l loys , the ex t rap ola t ion to B = 0 genera l ly pre-

sents no grea t p roble m as the spo ntaneou s effec l~ i smuch greater than the ord inary effec t except for d i lu tesamples a t low tem pera tures . Only in th is part icu larcase i s i t necessary to kno w how 6 p ( B , 0 ) varies withB an d 0 . The o rd ina ry mag netoresis tance of N i al loyshas been s tudied by Schwerer and Si lcox [9] w hoshowed that the varia t ions of p l l (8) and p l (B ) fo l low aKohler ' s law which d epends s t rongly on the type o fimpuri ty present .

For most o f the a l loys wi th which we wi l l be con .cerned , the ex t ra pola t ions are negl ig ib le . Except ionsare al loys, part icula rly NiCo, where residua l resistancesare low even a t qu i te h igh impuri ty con cent ra t ion s andwhere the ord inary magnetoresis tance is s t rong .

P,CB,--0)

O,(BL=O)r::~._---.O B.I.=OH=O

/Y

l .i g . 1 . R e s i s t iv i t y o f a f e r r o m a g n e t i c s a m p l e a s a f u n c t i o n o fa p p l i e d f i e l d f o r Hll i a n d H I i g e o m e t r i e s ( s c h e m a t i c ) .W i t h8 = H + 4 : r M a n d d i f f e r e n t d e m a g n e t i z i n g f a c t o r s f o r H l l a n dt tL, t h e s p o n t a n e o u s a n i s o t r o p y o f r e s is t iv i t y A p i s d e f i n e d asp lI (B I I = O ) - p l ( B j . = 0 ) , w h e r e t h e s e v a l u e s a r e o b t a i n e d b ye x t r a p o l a ti o n . N o t e t h a t Po(H = O ) d e p e n d s o n t h e m a g n e t i ch i s t o r y o f th e s a m p l e a n d i s g e n e r a l ly d i f f e r e n t f r o m ~ =1 [ p l l( 8 0 ) + 2 p I ( B I = 0 ) ] .

3 . E x per i m ent a l de t a il sWe prepared N i based samp les co n t a in ing d i f fe ren t

e lem ents a t var ious conc ent ra t ion s . These a l loys werep repa red by i nduc t i on me l t i ng i n a lumina c ruc ib le sor for the less resist ive ones (N/Co, NiFe and N iCu) byinduc t i on m e l ti ng on a co ld coppe r hea r th w i th weakl ev i t a t ion o f t he me l t . The a tmosphe re was a rgon o ra rgon /hyd rogen .

Th e a l loys were usual ly co ld ro l led in to the formof w ires or long p la tes , and were genera l ly annealed a t900 C under vacuum . Sam ple res is tances were m ea-sured a t 4 .2 K in appl ied f ie lds up to 30 kG ( in som ecases 70 k G) for para l le l and perpendicular or ien ta t ion sof the sample current re la t ive to the appl ied f ie ld . Asdefin ed in eq. (2) , the an iso t ro py AO/~ does not de-pend on t he samp le fo rm fac to r . The re a re, however ,possib i li t ies of error due to tex tur e or com pos i t ionp rob l ems .3.1. Problem of textures

The d i spe rsion o f t he exp e r imen ta l da t a on Ap/-~fo r N iC o a ll oys found in t he l i t e ra tu re [6 , 8 ,1 0 -1 3and our own resu l t s] drew our a t ten t ion to the pos-s ib i l i ty of tex tures in sam ples which we re supposed tobe rand om po lyc rys t a ll ine . From da ta on NiFe singlecrysta l s [14 ,15] i t can be shown that samples wi th cer-ta in tex ture s could have apparent Ap/~ values consid-e rab ly d i f fe ren t (by a fac to r o f up t o 2 ) f rom the va l-ue character i s t ic of a randomly orien ted polycrysta l -l ine sample of the same a l loy .

Som e N i based a l loys , par t icu lar ly NiFe and NiCo,are wel l know n for having d is t inc t tex tures w hich varywi th cast ing , co ld work a nd anneal ing procedures . Wefou nd a considerable spread of an iso t ropy values ford i ffer ent samples a ll com ing from a s ingle NiCo al loybu t wh ich had unde rgone d i f fe ren t trea tmen t s . W etheref ore X.ray analysed our samples af ter e tch ingand checking grain size. All our results refer to sampleswi th no p ronounced t ex tu r e , and wh ich we hope a rec lose to random polycry sta l l ine samples .

Fo r some samples we prod uced d i rec t ly cast rodsby suck ing t he mo l t en a l l oy i n to an Hum ina t ube ( see ,for exam ple , [16] . This process corresponds to a rap idque nch , and the gra ins grow inw ard rad ia l ly in a den-dri t ic form . X-rays show genera l ly random ly orien tedcryst rJ l ites . Agreem ent be tween these samples and


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O. Jaoul e t al. / Spontaneou s resist iv i ty anisotropy in Ni al loys 25T a b l e 1T h e r e s i d u a l r e s i s t . c i t y p e r p e r c e n t , t h e v a l u e s o f a = p o $ /P O 1' , t h e s a t u r a t i o n r e s i s ti v i t y a r t i s o t r o p y a p / ~ a n d t h e e s t i m z t e d a n -i s o t r o p i e s i n e a c h s p i n b a n d f o r N i b a s e d b i n a r y a l l o y s

I m p u r i t y p(O)l% a Ap lK (%) ( A p / p ) t ( ~ . ) (ZXp/p), (%)/ ~ 2 c m / % a t .I m p u r i t ie s w i t h n o V B SM n 0 . 7 9 * 1 1 t o 1 7 c 9 . 5 0 . 5 *F e 0 . 3 7 " 1 5 t o 2 3 c 1 9 . 5 1 *C o 0 . 1 6 " 2 3 t o 3 5 c 2 8 3 "P d 0 . 1 4 * ~ 3 * 2 . 0 0 . 5 aC u 0 . 7 7 * 8 t o 1 0 " 7 . 8 0 . 5 *Zn 1 .0 1 a . . -8 6 .5 aA ! 3 . 0 5 a ~ 6 4 . 8 aS i 3 . 6 a ~ 4 2 . 8 a

S n 3 . 5 a ~ 4 3 . 5 a

A u 0 . 6 2 * 6 t o 8 * 7 . 9 0 . 7 *

1 0 .5 0 . 5 - 1 . 2 0 . 4

2 0 2 * - 1 . 0 0 . 3 "3 0 3 * - 1 . 2 ~ 0 . 3 "

8 . 0 0 . 3 "

5 0 .5

- 2 . 3 1 . 5 "

I m p u r i t i es w i t h V B S

- 1 . 5 0 . 5

3 . 8 5 +- 0 . 3 - 1 . 2 0 . 68 . 0 0 . 5 - 2 . 5 1 . 3

V 5 . 0 * 0 . 5 5 t o 0 . 6 5 c 0 . 6 *C r 4 . 8 * 9 . 3 5 t o 0 . 5 c - 0 . 2 8 *N b 5 . 3 b 0 . 5 b + 0 . 1 5 *M o 6 . 0 b 0 . 3 5 * + 0 . 0 5 *R u 5 . 2 6 b 0 . 1 4 * - 0 . 8 2 *R h 1 . 81 b 0 . 3 ~ + 0 . 0 5 *W 4 . 9 b 0 . 5 0 * + 0 . 8 aR e 6 . 2 b 0 . 1 8 * - 0 . 4 5 *O s 5 . 9 b 0 . 1 2 * + 0 . 2 5 *I r 3 . 9 b 0 . 1 7 * - 1 . 5 2 *Pt 0 .8 b 0 .2 1 3 * + 0 .4 *

+ 4 . 4 2 " - 2 . 5 1 2 "

+ 1 . 2 6 " - 0 . 7 + - 0 .2 "

0 . 5 0 . 5 - 1 . 0 9 . 3+ 3 . 7 1 - 1 . 5 0 . 4

+ 4 2 - - 1. 5 0 . 5+ 9 3 " - 0 . 7 5 + - 0 . 3 "- 4 2 " - 1 . 0 0 . 3 "+ 4 1 .5 - 0 . 3 0 . 3

N o t e : ( a ) R e f . [ 1 1 ] , ( b ) r e f . [ 3 0 ] , ( c ) r e f . [ 5 ] , a n d t h o s e m a r k e d * a r e f r o m t h e p r e s e n t w o r k a n d r e f . [ 1 5 ] . I n t h e l a s t t w o c o l u n m s ,t h e v a i u e s n o t m a r k e d w i t h * w e r e o b t a i n e d f r o m a r e a n a l y s i s o f t h e r e s u l ts o f r e f . [ 8 ] i n o rd e ~ to s h o w t h e i n f l u e n c e o f t h ec h o i c e o f t h e p r e s e n t a v a l ue s . It s h o u l d b e n o t e d t h a t t h e u n c e r t a i n t i e s i n v o l v e d a r e m u c h g r e a te r l h a n i n t h e p r e c e e d t n gc o l u m n s .


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26 O. .laoul et al. / Spontaneous resistivity aniso trop y in At" alloysc onve n t iona l l y p r e pa r e d sa mples w a s good .

T he on ly t r u ly s a t i s f a c to r y w a y t o ove r c om e thet e x tu r e p r ob l e m w ould be t o ma ke me a sur e me nt s onse ts of monocrys ta ls and take t i le cor rec t polycrys ta l -l ine a , ,emge f rom these va lues . However , th i s would bea n e x t r e me ly t e d ious p r oc e dur e .3. _." Parasitic impurities

We used 5 N or 6 N meta ls and t i le pr inc ipa l pol lu-t i on c ome s f r om the m e l t i ng in t he a lumina c r uc ib l e sand the annea l ing in quar tz tubes . Pr inc ipa l r es idua lparas i t ic impur i t ies a re AI and Si . A sample of pure Niprepared in the same way as the a l loys gave a res idua lresist ivi ty rat io (R R R ) "-- 800 .

4 . Exper imenta l r esul t sO ur e xpe r ime n ta l r e su lt s f o r b ina ry sa mple s a t 4 .2 K

toge the r w i th some r e su l t s by o the r a u thor s a t l ow t e m-pera ture a re given in table I . F or each a l loy se r ies N i Xthe re i s an anisot ro py va lue (Ap/~-) which i s charac-te r i s t ic of the impur i ty X. For most impur i t ies th is an-i sot ropy i s easy to de te rm ine , as we f ind l i t t le con -cent ra t io~; dependence a t concent ra t ions of the order1% and gc~od agreemen t w i th va lues repor ted by other

20

15

I,1,

ia t

o o , o

I

~ %, , o ~

o 20 40 60 80Impur ty concent rot ion, o t %F i~ . 2 . C on c en t ra t i on dependenc e o f t he r es i s ti v i ty an i s o t r opyfo r t h ree N i bas ed s e r ies s how ing s l rong e f f ec ts a t T = 4 .2 K .A N iCo, =~NiFe, NiC u. Crosses and fu l l symbols : presentresults. Fmp ty symbols: 3mit [6].

a u thor s w he r e t hese e x i s t. H ow e ve r , t he c a se o f im-pur i t ie s w hic h ha ve bo th h igh a n i so t rop i e s a nd l o wspec i f ic res i s t iv i t ies a re more compl ica ted; we show thede t a i l s o f t he c onc e n t r a t i on de pe nde nc e o f t he a n .i s o t r o p y f o r N i C o , N i F e a nd N iCuin f i gs . 2 a nd 3 . ForN i C o our resul t s a re in reasonable agreement wi ththose o f D or le i jn a nd M ie de ma [ 8 ] ove r t he c onc e n-t r a t i on r a nge up t o 5% w hic h t he y i nve s t iga t e d , bu ta r e c ons ide r a b ly a bove t hose o f Smi t [ 6 ] a nd M c G ui r e[13] (except tha t McGuire a l so f inds qui te a high an-i so t r opy f o r a N i C o 30% a l l oy) . For N i F e a nd N i C ut he r e i s muc h be t t e r a g r e e me nt be tw e e n t be r e su l t sf r om d i f f e r e n t g r oups . T he d i sa gr e e me nt f o r N i C o ca npa r t i a l ly be a sc ri be d to t he i n f l ue nc e o f unc on t r o l l e dt e x tu r e , bu t t he e f fe c t o f pa r a s i t ic imp ur i t ie s i n t hesa mple s g iv ing l ow va lue s i s p r oba b ly a lso im por t a n t .As the res idua l r es i st iv i ty of NiC o i s par t icula r ly lo w,these a l loys a re ext remely sens i t ive to the presenceof unc on t r o l l e d impur i t i e s suc h a s W in t r oduc e ddur ing a r c m e l t ing .

i t can be seen f rom f ig . 3 tha t for each of thesethree a l loy se r ies there i s a cont inuous var ia t ion ofthe a n i so t r opy w i th c onc e n t r a t i on un t i l a p l a t e a u i sr e a c he d . We ha ve a l r ea dy po in t e d ou t [ 17] t ha t t hec ond i t i on f o r s a tu r a t ion a ppe a r s t o be t ha t t he r e s id -ua l r es i s t iv i ty of the a l loy must be grea te r than about

a o A , l s i % . . . . G

2 5 Ni Co

,o. / .

t o l o - 2

5 [," " 5 i / 5 t / N C u2 4 6 " ~ ' 2 4 ~ 2 4 8 1 0

0 " Conc---c-~ a t on , %F ig . 3 . C onc en t ra t i on dependenc e o f t he r es i s t i v i t y an i s o t r opyfo r NiCo , N iFe and N;Cu i n t he l ow c onc en t ra t i on r eg ion .N iC o : A p res en t r es u l t s , A D or le i j n and M iedem a [8J ,VM cGuire [13 ], n Smit (6).Nil:e: present results, o Dorleijn and Miedema [81,@ Sm it[61.N/Cu: x pre sen t results, +Stair [61; Dorleijn and M iedema [8 ],In each case, arrows indicate the co ncentration where P0 =2 ufZ cm .


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O. Jaoul e t ai . / Spo ntaneous res is t iv i ty anisotropy in Ni al loys 272 #12cm . T he conc en t ra t ion co r respond ing to th is c ri -ter ion is indic ated for each case in f ig . 3 . Fo r Fe andCu the an i so t ropy r em ains unchanged f rom th i s con-cen t ra t ion up to ab ou t 25%, and fo r Co up to ab ou t50%.

T he in i ti a l approach to the p la teau va lue appearsas an in tr ins ic ef fec t ra ther tha n, for in s tanc e, a grad-ua l change o f chara c te r o f the im p ur i ty w i th concen-t r a t ion , because fo r each o f the th ree a l loy s e ries theanisotropy increases regular ly and in the same fashionwi th conce n t ra t ion un t i l the r es idua l r es is t iv i tyreaches abou t 2 /a ~c m . W e have d i scus sed th i s e ff ec tand as soc ia ted concen t ra t ion de pen den t changes inthe deviat ions f rom Matthiessens rule , in the ordinaryHall e f f ec t and in the o rd inary m agne to res i s tance [17].

For the v ar ious al loy ser ies the p latea u value ofAp/p pers is ts unt i l a drop is observed only at concen-t r a t ions so h igh tha t in te rac t ions be tw een im pur i t i esa re ce r ta in ly im por tan t . I t seem s tha t Ap[p is onlysens i t ive to qui te major changes in the al loy s t ructure.Thus for ins tance for NiCu a l loys the an i so t ro py be-gins to be af fected by the change in electronic s t ruc-tu re and the low er ing o f T ~ on ly when the co ncen t ra -t ion increases above about 25% (cf . calculat ions byBrouers et al. [18]) and for NiFe al loys Ap/~ fal lsd ram at ica lly at the concen t ra t ion co r respond ing tothe Ni 3 Fe com pou nd . In NiCo a s imilar drop occursabove 50% concen t ra t ion .

W e shou ld perhaps po in t ou t tha t the an i so t rop ieswe obtain for cer tain NiCo samples are the highes tever repor ted on any sys tem up to now. This sugges tspo:~s ible techn ologic al ap pl icat ions , but for these sam-p les a t room tem pera tu re the an i so t ropy has d roppedto 6%, which i s l i t t l e m ore than the a n i so t rop y fo rNiFe near the perm al loy concen t ra t ion .

5 . Ternary al loysW e have a l so inade m easurem en ts on a num ber o f

te rnary a l loys, N iCuCr , N iCoV, N iFe l r , N iFeO s ( fig. 4 ).Res idual res is tance measurements on ternary al loyscan be analysed to give the values of a = Po $/Po t fo rthe im pur i t i es p resen t [ 1 -5 ] .

S im i la r ly , spon tan eous an i so t ropy r esu l t s on s e t s o fternary al loys can be used to obtain the anisotropieswith in each sp in subband (Ap/p) t and (Ap/p)$ sep-arately for each im pu rity . This idea has been exploited

ap t? "1.

Ni O s F e

o [ , 0 . ' 2 ' o 1 4 ' o : 6 '.1

-2Ni Ir F e d

Fig. 4.The residual resistivity anisotropy of ternary Ni basedalloys Ni I _ x _ y F e x B y w i t h B = It, Os as a function of therelative concentration t = x / x + y . The curves are fits usingthe technique described in the Appendix and correspond tothe parameters for Fe, lr and Os given in table 1.

by Dor!ei jn and Miederr , a [8] who o btain ed es t ima tesfor a num ber o f imp uri t ie s in Ni . A!' :!hough this w orkis in teres t ing, we do n ot cons ider i t def ini tive forthree main reasons"(1 ) Fo r a num ber o f the s am ples m easured in th i swo rk, tae res idual res is t iv i ty is signif icant ly higherthan values obtained by ourselves or by other groups[2, 19] . This sugges ts inadver tent in troduct ion ofparas i t ic impuri t ies in to the al loys , which could ser .ious ly a f f ec t Ap /~ .

(2) ~ ,ome of their te rnar y al loys do not seem to besuf f ic ien t ly conce n t ra ted ( espec ia lly those con ta in ingCo) to avo id the low con cen t ra t ion e f f ec t desc r ibedb e f o r e .

(3) The final set of t~ values [8] differs considerablyf rom m os t o f the va lues es tim ated f rom ind iv idua l se t sof te rnary al loys , us ing the same samples [4] . This ap-pear s to be because a g loba l com p ute r t r ea tm e n t i snot the ideal techniq ue for analys ing; the data . W e havediscussed the tern ary al loy res idual res is t iv i ty resul ts[5], and we find a set of final "best 'values" for a dif-f e ren t f rom those o f r e f . [8 ] bu t com pat ib le w i th theexper im en ta l da ta o f r e f . (4 ) and in ag reem e nt w i thprevious work [20] .

T he m ethod w e have used fo r de te rm in ing (Ap/~ ) l '


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28 O . . l a o t d e t a t . / 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 ) , i n N i a l l o y sa n d (Ap/p)$ values f rom ternary a l loy ~ . ~ o / ~ d a t a w h e nthe a values are known i s d i scussed in the appendix .We have appl ied th i s analys i s to our t ernary a l loy datao n N i C o V , N i C u C r a n d N i F e i r , N i F e O s a l l o y s a n d t othe data of ref . [8] . Values , wi th associa ted uncer-ta in t ies , are g iven in t ab le 1 , a l though the roug h qual -i t a t ive conclus ions are s imi lar to those that can bedrawn f rom the equivalen t t ab le in ref . [8] , the ind i -v i d u al p a ram et e~ a re s en si ti v e to t h e ch o i ce o f t h e avalues.

i t s h o u ld b e n o t ed t h a t (Ap / p )~ i s eq u a l t o - -1 %wi t h i n t h e ex p er i m en t a l i n ce r t a in t y fo r a ll t h e i m -p u r i t ie s in t h e t ab l e (w i t h t h e p o s s ib l e ex cep t i o n o fP t~. On the o ther ha nd , for im puri t i es wi th 0 t greatert h an I (A p / p ) t v a r ie s f ro m +3 0 % t o +4 % an d is c l o se l yp ro p o r t i o n a l t o t h e v a l u e o f a , wh i l e fo r i m p u r i t i e swi th a l ess than I (Ap/p}? v ari es f ro m +9 % t o -4 %and seems unrela ted to c t. (We do no~t agree w i th thes ta tement in ref . [8] that the (Ap/p)t values are ap-p ro x i m a t e l y t h e s am e fo r a l l i m p u r i t ie s . )

Ex p l an a t i o n s fo r t h i s b eh av i o u r w i l l b e d i s cu ss ed i nthe next sect ions .

6 . Bas i s of the model for the res i s t iv i ty an i so t ropyQui te general ly , in a fer romagnet ic metal such asNi sp in ? and sp in $ e lect rons carry cu rren t in paral le l .

~,~,'e can defin e at T = 0 intr a-s pin resid ual resis tivitiesPo ~ a: ld Pc, ~. Fo r a given im pur i ty in N i the rat ioa = p 0 , t / p o t i s e s t i m a t ed b y m eas u rem en t s o n t h e t em -pera ture dep ende nce of b inary a l loy ~res is tivi ti es or bymeasurements on ternary a l loy res idual res i s t iv i t i es(see . e .g . [5]). In t ab le 1 , we l i st the values o f p o pera t o m i c p e rcen t , o f a v a l u es an d o f Ap / ~ fo r d i f f e ren timpuri t i es in Nf . We wi l l d i scuss the in terpreta t ion oft h es e v alu es i n t e rm s o f t h e w e l l -k n o wn s - d m o d e l .6.1. The s -d model

I . , t h i s - " ' ~ - ', , , , , u ~ = t , the host meta~. . . . . is r ep res en t ed a shaving a t igh t b inding d-band and a f ree-elect ron- l ikeco n d u c t i o n b an d . F o r t h e f e r ro m ag n e t i c m e t a l s , t h ed-band i s po lar ized bu t the s -band can be tak en forpresent purposes as unpolar ized . A t r~msi t ion impuri tyis screened essent ia l ly by the d-elect rons . Nick el i s as t ro n g f e r ro m ag n e t . Th e d ? e l ec t ro n s li e co m p l e t e l yb e l o w t h e F e rm i en e rg y , wh i l e t he d $ b an d h as a r e l-

~ c m

~IiFe

10 20I m p u r l l y c o n c e n t r Q l i o n ( < a t .%)

i . . . . . . .3 r , ' o 5 ' 0Fig. 5. Residu al resistivity as a func tion of conc entration forNiCo and Nil:e.

a t i v e l y h i g h d en s i t y o f s ta t e s a t t h e F e rm i en e rg y .F r o m m e a s u re m e n t s o f th e m a g n et ic m o m e n t o n

an d a ro u n d t h e i m p u r i t y s i t e ( s ee , fo r ex am p l e , Lo w[21 ] ) an d u s i n g t h e p i c t u re d ev e l o p ed b y F r i ed e l [2 2 ] ,K a n a m o r i [ 2 3] a nd C a m p b e l l a n d G o m e s [ 2 4 , 2 5 ] ,o n e co n c l u d es t h a t :

(1 ) F o r s o m e t r an s i t i o n i m p u r i t i e s i n Ni ( M n , F e , C o ,C u , P d , A u ) , t h e s c reen in g i s d o n e en t i r e l y b y t h e d ~b an d s o t h a t t h e re a re n o d t s t a t e s a t t h e F e rm i l ev e l .F o r s - p i m p u r i t ie s , t h e p o t en t i a l i s a t t r ac ti v e s o ag a int h e re a re n o d t s t a t e s a t t h e F e rm i en e rg y

(2 ) F o r o t h e r i m p u r i t i e s (C r , R h , R u , I r , Os , R e) as p i n t v i r tu a l b o u n d s t a t e (VB S ) o v e rl ap s t h e F e rm ien e rg y , f ig . 8 . Th i s VB S co r res p o n d s t o a r e s o n an t ds t a t e w i t h m o re o r l e s s s t ro n g s -d h y b r i d i za t i o n g i v i n gi t an en e rg y wi d t h A . M an y ex p er i m en t a l r e s u l t s o nthe res i s t iv it i es [20] , therm oele ct r ic pow ,~r [26] , e lec-t ronic sp eci f ic heat [27] , and Hal l ef fect [15] of sucha l l o y s g i ve i n fo rm a t i o n ab o u t t h i s VB S .

(3) F inal ly , for s t ron gly repuls ive imp~ar it ies a t theb eg i n n i n g o f a t r a n m i o n s e r ie s , t h e re is f~ d rl y s t ro n gs c reen i n g wi t h co n s i d e rab l e s - -d m i x i n g fo r b o t h d i r ec -t i o n s o f s p i n .

7 . R es i s t i v i t y an i s o t ro p y m ech an i s m sI f we as s u m e t h e s p i n -o rb i t e f f ec t is a weak p e r -

t u rb a t i o n o n t h e e l ec t ro n i c s t ru c t u re o f t h e m e t a l ,t h en t h e l o wes t o rd e r m ech an i s m s wh i ch can l ead t oa s p o n t an eo u s r e s i s t i v i t y an i s o t ro p y a re s eco n d o rd e r


7/12

o. Jaou l et al. ] Spontaneous resistivity anisotropy in Ni alloys 29i n t h e s p i n - o r b i t i n t e r a c t i o n * . W e w i l l d e s c r i b e t h e s el o w e st o r d e r m e c h a n i s m s w i t h i n t h e s - d m o d e l .

7 .1 . S p i n m i x i n g o r S m i t m e c h a n i s m ( L - S + )U s in g t h e s - d f r a m e w o r k , t h e m o s t i m p o r t a n t m e -

c h a n i s m l e a d i n g t o s p o n t a n e o u s m a g n e t o r e s i s t a n c e h a sa l re a d y b e e n p r o p o s e d a n d t e s te d e x p e r i m e n t a l l y [ 6 , 7 ] .I n t h e p r e se n c e o f s p i n - o r b i t c o u p l i n g i n t h e d - b an d ,a c e r ta i n a m o u n t o f d t c h a r a c t e r is m i x e d i n t o t h e d $s ta t e s a t t h e F e r m i e n e r g y t h r o u g h t h e i n f l u e n c e o f t h eL - S * o p e r a t o r . T h i s p e rm i t s s s c a t te r i n g i n t o t h e d $p a r t o f t h e m a i n l y d $ F e r m i su r f ac e . T h e d t - d $ m i x i n gis n o t i s o t r o p i c b e c a u s e t h e m a g n e t i z a t i o n d i r e c t io np r o v i d e s a p r i v i l e g e d a xi.s f o r t h e s p i n - o r b i t p e r t u r b a -t i o n , s o i t p r o v i d e s a m e c h a n i s m f o r t h e r e s i s ta n c e a n -i s o t ro p y . T h e a n i s o t r o p y m a y b e s e e n a s a n a n i s o t r o p ict r a n s f e r o f a c e r t a i n a m o u n t o f s p i n ,I, r e s i s ti v i t y i n t ot h e s p in t b a n d . O n t h i s m o d e l , a t l o w t e m p e r a t u r e

I

. -L _ . .. .. .. .. . I . . . . .

10 20 3 0 o tFig. 6 . The charac ter is t ic va lue of A p / ~ a t 4.2 K for var iousimpu ri t ies in Ni as a fu nct io n of ~ : : pO/p~ (see tex t and tab-le l). The straight line is & p/ ~ = 0.01 (~ - 1) .

Ap/-fi = 3,(a - 1 ) , ( 3 )w h e r e a = P o $ / P o t a n d 7 is a c o n s t a n t a p p r e x i m a t e l ye q u a l t o ( k / H e x ) 2 a n d w h i c h c a n b e e s t i m a t e d a p r i o r ia s b e i n g a b o u t 1 0 - 2 . k is t h e s p i n - o r b i t a v e ra g e v a lu ei n t h e s p i n $ b a n d o f n i c k e l , a n d H e x i s t h e e x c h a n g ee n e r g y w h i c h s p l i t s t h e d t a n d d $ b a n d s .

W e c a n c o m p a r e t h e Ap/-~ c h a r a c t e r i s ti c o f e a ch N i Xs y s t e m w i t h t h e e t v a l u e s , fi g. 6 . E q . ( 3 ) p r o v i d e s a g o o df i t t o e x p e r i m e n t w i t h 3 ' ~ 0 . 0 1 ( t h i s v a l u e i s s l i g h tl yd i f f er e n t f r o m t h e o n e a d o p t e d b y C a m p b e l l e t a l. [ 7 ]m a i n l y b e c a u s e o f t h e r e vi s ed e s t i m a t e s o f t h e c h a r -a c t e r i s t i c A p/ - ~ v a l u e s f o r N i C o a n d N i F e ) .

T h e s p i n t b a n d f o r a l l o y s s u c h a s N i C o o r N i F e isa l m o s t p u r e l y o f sp c o n d u c t i o n c h a r a c t e r a n d s o s h o u l db e a l m o s t u n i n f l u e n c e d b y t h e s p in - o r b i t i n t e r a c t i o n ,t h u s g i v in g n o r e s i s t i v i ty a n i s o t r o p y . T h e f a c t t h a t i np r a c t i c e t h e l a rg e s t a n i s o t r o p i e s a r e o b s e r v e d f o r p r e -c i s e ly thoo ~ l l c ,y~ , ,,h~, -, - ,h ,~ , ,~ s p * . . . s p in t " ' . . . . . . . .lUtd 1 .1 UII~a r e c a r r y i n g a l m o s t a l l t h e c u r r e n t s h o w s w i t h o u t a n yd o u b t t h a t t h e d o m i n a n t e f f e c t m u s t b e d u e t o a S m i tt y p e o f m e c h a n i s m w h e r e th e s p i n - o r b i t a c ts to m i xs p i n t c h a r a c t e r i n t o t h e s p in $ d b a n d . T h e g o o d a g r e e -* The re l a tion be twe e n the o rder o f t he pe r tu rba t ion a nd the.symmetry of the phys ica l e ffec t was pointed out to us by

P. Lenglart.

m e n t b e t w e e n t h e e x p e r i m e n t a l r e s u l t s a n d e q . ( 3 }c o n f i r m s t ha ~t t h i s m e c h a n i s m d o m i n a t e r , , p a r t i c u l a r l yf o r i m p u r i t i e s h a v i n g a > 1 .

H o w e v e r , i f w e l o o k i n d e ta i l a t th e r e s u l t s f o r im -p u r i t i e s w i t h l o w v a l u e s o f A p / ~ 6 , f i g. 7 , t h e r e i s a n a p -p a r e n t l y r a n d o m s c a t t e r o f v a l ue s o f a b o u t - 19~ a r o u n dt ! ' c 7 ( a - - 1 ) l i n e. F o r t h e s e i m p u r i t i e s t h i s s c a t t e r ism u c h g r e a t e r t h a n t h e e x p e r i m e n t a l e r r o r s . I t i s o b -

00/ M ---~do R h . ~ ~.5

_l Ru

Fig. 7 . An enlargeme nt of the par t of f ig . 6 correspo nding toimpu ri t ies wi th low res is t iv i ty anisotropies . The l ineAp /ff = 0 .01 (a - - 1) (Sm it m echanism) does not f i t the re -suits w ell for alloys with ot < 1.


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3 0 O . d a o u i e i a l. / S p o n t a n e o u s r e s i s ti v i ty a n i s o t r o p y i n N i a l l o y s

, d o u s t h a t a is n o t t h e o n l y r e le v a n t p a r a m e t e r w h i c hde te r m ine s ..k0f~ i. Th e de t a i l s o f t he e l e c t r on ic s t r uc -t u r e o f t h e i m p u r i t y a r e i m p o r t a n t . I n t h e n e x t s e c t i o n ,we wi l l d i sc uss a d i f f e r e n t m e c ha n i sm in o r de r t o e x -p la in the se de v ia t ions .Z Z l n t ra b a n d o r L = S , m e c h a n i s m

S o t a r we ha ve ignor e d the L z S z t e r m i n t h e s p i n -o r b i t i n t e r a c t ion . Howe ve r , t h i s t e r m c a n a l so l e a d toa n a n i so t r op ic r e s i s t i v i ty by l i f t i ng the de ge ne r a c y o fthe d s t a t e s f o r a g ive n d i r e c t ion o f sp in . B e c a use thesc a t t e r ing in to e a c h rl s t a t e t a ke n se pa r a t e ly i s a n i so -t r o p i c , t h i s l if t in g o f d e g e n e r a c y l e a d s t o a n i s o t r o p i cres is t iv i ty .

S uppo se th e r e i s a s e t o f de ge n e r a t e d 1' VB S ne a rt h e F e r m i le v e l, fi g. 8 ( a ) . W h e n w e i n t r o d u c e s p i n - o r -b i t , t h e d i f f e r e n t d s t a t e s w i l l sp l i t, f ig . 8 ( b ) , ( c ) . F o rthe c a se o f f ig . 8 ( c ) , t he r e wi l l be r e l a t ive ly m o r e I 0)s t a t e s t h a n t h e o t h e r s a t t h e F e r m i le v e l. F o r f ig . 8 ( b ) ,t he r e wi l l be r e l a t ive ly m o r e 12) s t a t e s . Now , f o r as p h e r i ca l s c a t t er i n g p o t e n t i a l , a f r e e e l e c t r o n w i t h kve c to r pa r a l l e l t o t he m a gne t i z a t ion c a n on ly be sc a t -t e r e d i n t o I m = 0 ) s ta t e w h e r e a s a n e l e c t r o n w i t h kve c to r pe r pe nd ic u la r t o the m a gne t i z a t ion i s s c a t t e r e din to a s t a t e{ v - , - v - ) ( ~ - ~ + I . . .. ' . } ) + ~ I 0 ) .The s i tua t ion o f f ig . 8 ( c ) w i l l f a vour the sc a t t e r ing o fthe e l e c t r on pa r a l l e l t o t he m a gne t i z a t ion g iv ingp , > p , whe r e a s the s i t ua t ion o f f ig . 8 ( b ) w i l l l e a d top~ < p~.. We can ca lcul a te th e s ize of th e e f fec t to bee x p e c t e d f r o m t h i s m e c h a n i s m .

n(E); EFd ', l " X .E~ E

a )

n (oE)~ EF n( E )l EF

6n(E )~ b) c)f i g . 811 ( a ) S c h e m a t i c p i c t u r e o f a v i r t u a l b o u n d s t a t e n e a r t h es p i n r F e r m i s u r f a c e. ( b ) , ( c ) l o c a l s p i n - o r b i t c o u p l i n g o n t h ei m p u r i t t y s i t e l i f t s t h e , d e g e n e r a c y o f t h e d i f f e r e n t m z s t a t e sm a k i n g u p t h e V B S . & p / ~ w i ll d e p e n d o n t h e p o s i t i o n o f E Ow i t h r e s p e c t t o t h e F e r m i e n e r g y E l .. .

The 5d 1 ' r e sona n t s t a t e s a r e a s sum e d to ha ve e ne r .giesb . ~ t : E~+ ~X'rn ,w h e r e f i ~ t i s t h e e n e r g y o f t h e d i m p u r i t y l e v el s u b -m i t t e d t o a l o c a l s p i n - o r b i t c o u p l i n g ~ , ', ( - - 2 < m < + 2 ) .E a c h o f t h e s e s ta t e s is t h e n m i x e d w i t h a n l = 2 p a r to f t h e p l a ne wa ve s t a t e s , l e a d ing to a pha se sh i f t 11~ tg ive n byta n r ~ n t = A / ( E ~ t - E F ) , ( 6 )w h e r e A i s th e v i r t u a l b o u n d s t a t e w i d t h a n d E F t h eF e r m i e n e r g y .

T h e n , t h e s c a t t e r i n g a m p l i t u d e d e v e l o p ed i n s p h e r -i ca l h a r m o n i c s f ( 0 , O )

o~ +!4,r ~ ~ (ein _ l )f ( o , ) = z = o , , , = - zx Y ; " (O kOk ) Y (Ok , % ) ( 7 )

m a y b e d i r e c t ly c a l c u l a t e d f r o m e q s . ( 5 ) a n d ( 6 ) a n dp u t i n t h e e x p r e s s i o n o f t h e d i a g o n a l p a r t o f t h e s c a t -t e r ing c r oss se c t ion t e nso r~ Ou u= ~ 2 I (O k - -O k ' )U l 2 d ~ g d ~ k ' , ( 8)

I (o k - O k ,)U l 2 d ~ 2 k d ~ k ,w h e r e k a n d k ' d e n o t e t h e i n c i d e n t a n d s c a t t e r e dwa ve ve c to r s , r e spe c t ive ly , i n d i r e c t ions ( 0g0k) , ( 0g ,0 g ' ) w i t h r e s p ec t t o t h e m a g n e t i z a t i o n d i r e c t i o n , u i s au n i t v e c t o r i n t h e d i r e c t i o n o f t h e c u r r e n t .

R a p i d l y t h e c a lc u l a t i o n o f Ouu s h o w s t h a t t h e f i v ed pha se sh i f t s r/~ a r e we igh te d ve r y d i f f e r e n t ly . Thes y m m e t r y b r e a k d o w n i s s u f f i c i e n t t o av o i d c a l c u l a t i o non o the r pa r t i a l wa ve s tha n l = 2 a nd a l so a n a ve r a g ingof the c r os s se c tions ove r a ll t he k ' s c a t t e r e d wa vev e c t o r s .

T h e n , w i t h i n t hi s s i m p l i f i c a t i o n - o t h e r w i s e t h e r e -s t i lt doe s no t d i f f e r g r e a t ly - t he tw o c r oss se c t ion sc a n b e w r i t t e n

Oztz cc I/' ]2 cos 2 0 s in 0 dO dO ,

r o : I jq2 s in 2 0 cos 2 0 dO d 0 ,O x x ( 9 )


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O. Jaoul et al. / Spon taneou s resistivity anisotropy in/~ i alloys 31w h e r e z a n d x d e n o t e d i r e c t i o n s p a r a ll e l a n d p e r p e n -d i c u l a r t o t h e m a g n e t i z a t i o n .

T h e a n i s o t r o p y o f t h e c r os s s e c t i o n t e n s o r c o m e sf r o m t h e f a c t t h a t t h e s c a t t e r i n g a m p l i t u d e f ( O , ~ ) isa n i s o t r o p i c , l i k e t h e I ( kl d m ) l 2 i n t e g r a l a s w e s h o w e d i nt h e s i m p l e e x a m p l e a b o v e . F i n a l l y , t h i s m e c h a n i s m i n -d u c e s a n i n t r a - s p i n t r a n s f e r o f r e s i s t i v i t i e s f r o m p a r a l l e l(11 o r z z ) t o p e r p e n d i c u l a r ( 1 o r x x ) d i r e c t i o n s o f t h ec u r r e n t w i t h r e s p e c t t o t h e m a g n e t i z a t i o n . T h e np ~ = ( 1 + 2 / 3 ) P~o + 2 7 p ~ ,

p ~ = ( l - 2 " y ) p ~ , P I = ( I - 3 ' ) P ~ , ( 1 0 )w h e r e p ~ a n d p ro a r e r e s i s ti v i ti e s w i t h o u t s p i n o r b i ta n d 7 is t h e S m i t m e c h a n i s m p a r a m e t e r . W e f in d~ = ~ (r lt2 K ) = A r ( r l ~ , K ) / s i n 2 r l , ( 1 1 )w i t hY ( , f i , K ) - 2 , r i o + s i . 2

_ 2 ( s in : ~ r / t 2 + s i n 2 ~ - 2 ) ( 1 2 )a n dp r o s s i, , r i o , ( 1 3 )w h e r e y ( ~ , K ) is a f u n c t i o n o f t h e p h a s e s h if t ~ a n dis r e l a te d t o t h e p o s i t i o n i n e n e r g y o f t h e V B S r e l a ti v e l yt o t h e F e r m i l e v e l , a n d d e p e n d s a l s o o n t h e l o c a l s p i n -o r b i t c o u p l i n g ~ ' a n d t h e w i d t h A . A i s c o n n e c t e d w i t ilt h e s - d h y b r i d i z a t i o n a r o u n d t h e i m p u r i t y p o t e n ti a l .M o r e p r e c i s e l y , Y d e p e n d s o n t h e r a t i o K = k ' / A .

F i n a l l y , t h e r e s i s t i v i t y a n i s o t r o p y m a y b e r e w r i t t e na s

( A p / p ) t ='y(x + 3/3; ( A p / p ) $ = - , y ( 1 4 )o r

A p / p = ' r ( a - 1 ) + 3 [ a / ( a + 1 )] /3 . ( t 5 )W h e n X ' < < A ( fi r s t s e ri e s t r a n s i t i o n i m p u r i t i e s )

7/3 = - ~ ( ~ , ' / A ) 2 ( 1 - - 4 c o s Z r / t : ) s in 2 r ~ . ( 1 6 )

/ K=.3

- ' ~ / 20Fig. 9. The f unction Y(r/O~ , K) representing the calcula tedL z S z effect anisot ropy for the spin ~ resonant d i f fusion, l 'ora given impurity n ot can be evaluated from the resist ivity oO-K = MA, wi l l be equal to ab out 0 .3 , 0 .5 and I for f i rs t, secondand th i rd ser ies t ransi t ion impuri t ies in Ni .

T h e a p p r o x i m a t i o n ( ~ , '/ A ) 2 < < 1 is n o t v a li d f o r s e c -o n d a n d t h i r d s e r i e s i m p u r i t i e s .

T o o b t a i n n u m e r i c a l e s t i m a t e s , w e t a k e A = O . 1 5 e V ,0 . 2 5 e V , 0 .5 e V f o r V B S w i d t h s o f f i rs t , s e c o n d a n dt h i r d s e r ie s im p u r i t ie s , r e s p e c t i v e ly ( v a l u e s e s t i m a t e df r o m s p e c i fi c h e a t [ 2 7 ] a n d t h e r m o e l e c t r i c p o w e r [ 2 6 ] * ).

F o r t h e l o c a l s p i n - o r b i t c o u p l i n g ;~ ', w e t a k e t h ea t o m i c v a l u es 4 X 1 0 - 2 , 0 . 1 2 a n d 0 .5 e V , r e s p e c t i v e l y .f o r t h e t h r e e s e r ie s o f e l e m e n t s . L J si ng t h e s e p a r a m -e t e r s w e e s t i m a t e L z S z a n i s o t r o p i e s a s s h o w n i n f ig . 9 ,w h e r e w e p l o t t h e f u n c t i o n Y (r~ , X ' / A ) a s a f u n c t i o no f r / ~ f o r v a r i o u s v a l u e s o f X ' / A . T h e s i m p l e e x p l a n a -t i o n g i v e n in i t i a ll y is s t i ll 'v a li d q u a l i t a t i v e l y O > P :

( i . e . w h e n t h e V B S i s c e n t r e do r p h a s e s h i f t s n e a r g r rn e a r t h e F e r m i l e v e l ) a n d O il < P t f o r n e a r l y f u l l o rn e a r l y e m p t y V B S .

A n e x a c t l y e q u i v a l e n t t e r m s h o u l d e x i s t i n t h es p i n ~, b a n d , b u t c a n b e e x p e c t e d t o b e m u c h w e a k e r ,a s i n t h e f a c t o r ( X ' / , ~ ) :' t h e V B S w id tl '~ A w o u l d b e r e -p l a c e d b y a n e n e r g y o f t h e o r d e r o f th e d b a n d w i d t h .D e v i a t i o n s f r o m t h e S m i t t e r m c a n b e e x p e c t e d t o b es m a l l f o r t h e s p i n ,I, b a n d . W e c a n c o m p a r e t h e p r e -d i c t i o n s o f t h is m o d i f i e d m o d e l w i t h th e e x p e r i m e n t s ,s e c t i o n s 4 a n d 5 a n d t a b l e 1 .

F i r s t , f o r n o i m p u r i t y d o e s t h e v a l u e o f ( A o / p ) ~d e v i a t e g r e a t l y f r o m - 1 % , t h a t i s t o s a y ( A p / p ) $ ~ 7i n a g r e e m e n t w i t h e q . ( 1 4 ) . F o r s p i n l , f i r s t l y ( A p / f , ) t

7 a f o r a l l i m p u r i t i e s w h e r e t h e r e i s n o V B S i n t h es p i n I ' b a n d ( M n , F e , C o , C u , A u , S n , A I , P d ) . T h i s i s

N o t i c e t h a t t h i s i n v o l v e s a s e c o n d p o w e r o f t h e s p i n -o r b i t i n t e r a c t i o n , i n a g r e e m e n t w i t h t h e g e n e r a l r u l e . * In fact , the VBS width should increase as the Z di f ferencebetween host and impuri ty increases.


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3 2 O . . l a o u i e t a i . / S p o n t a n c , g u s r c s i s t i r i t. v a n i s o t r o p y i n N i a l l o y s

s t ron~ suppo r t for t i~e dom inant inf luence of the Smi tmech:~,nism m the se cases. The rule fo r the tota l an-i sot ropy A ptp = 7(a - I ) i s well ob eye d.

On the other hand, for in lpur i t ies wi th VBS in thespin ? band, an ext ra /3 te rm must b ,e invoked. Thiste rm can be of e i ther s ign, and i s rough ly o f the orderof magni t ude es t im ated on the L z S z me c ha n i sm a bove .I I t~wever, i t is diff icu lt to recognize in the exp erim en-tal results the detai led var iat ion of the f l term as afunc t ion of r /~ predic ted by this model and shown infig. t ) .

The osc i l la t ing devia t ion f rom the Smi t va lueshown by the th i rd se r ies t r ans i t ion impur i t ies , f ig . 10,doe s se e m to show r ough ly t h i s be ha v iour bu t , a s w etake a very s impl i f ied descr ipt ion of the e lec t ronics~ruc ture of the V BS , we canno t rea l ly exp ec t prec iseagreement wi th the model . Never the less , for impur i t iessuch as I t , Os , Re , or Ru, the f i t be tween exper imenl ta land ca lcula ted va lues of Ap /~ (Sm i t + /3 te rm ) seemsto be quite go od bas ed on the r/~z valtues tak en fromtable 1 . Not ice tha t th i s type of impur i ty does not a~ ' -feet the spin ,L ba nd by extra sp in 4, scree ning charges.The main point i s tha t the devia t ion f rom the Smi t

6

~L !' I0 L : : ? , .,

L~ , , , I i 1i

Aut i ,. :. 10 . I xp er im en ta l res i s t i v i t y an i s o t r op i es l k) r t h i rd ser i esi m p u r i t i e s in N i p o i n t s . V a l u e s c a l c u l a t e d f r o m O . 0 1 ( ~ - 1 ):d a s h e d l in e . V a l u e s c a l c u l a t e d i n c l u d i n g I . ::S z m e c h a n i s m : f u lll i ne . ~ O v a l u e s u s e d i n t h e c a l c u l a t i o n a r e e s t i m a t e d f r o m r e -s i s ti v i ty v a l u e s o~ ). S i m i l a r v a l u e s w o u l d b c o b t a i n e d f r o mm a g . e l i c m o m , : n t d a t a , e x c e p t f o r t h e c a s e o f W { D u r a n d [ 31 )] ).

value is due m ainly to an e f fec t in the spin t VB S, andis s t ronges t for th i rd se r ies t r ans i t ion impur i t ies wherethe l oc al sp in - o r b i t c oup l ing i s s t r onge s t. T h i s i nd i-c a t e s t ha t t he o r i g in o f t he de v i a t i on f r om the Smi tva lue i s indeed th e L z S z e f f e c t i n t he V BS. H ow e ve r ,t he p r e d i c t ion f o r W i s ba d , so w e mus t be c a u t i ousa bo u t us ing t he de t a il e d qua n t i t a t i ve p r e d i c t i ons o ft h i s s imple mode l .

i t i s in te res t ing to remark tha t the L z S z t e r m c a nbe regarded as in t roduc ing a loca l e lec t ronic quadru-po l e mome nt on t he impur i t y s i t e , w h ic h g ive s t he im-pur i ty an anisot ropic c ross sec t ion (c f . Fr ieder ich andFe r t [28] who conside r the case of ra re ear ths) . T hise l e c t r on i c qua dr upo l e shou ld be r e l a t e d t o t he nuc l e a rquadrupole spl i t t ing observed a t th i rd se r ies t r ans i t ionimpur i ty s i tes in fe r romagnet ic hos ts ( [29] and re -fe rences there in) .

8 . Co n c l u s i o n

We have re f ined the model presented in an ear l ie ra r t ic le [7] to expla in the spontaneous res i s t iv i ty an-i so t r opy o f f e r roma gn e t i c me ta l s , a nd i n w hic h w eshow e d tha t t he Sm i t sp in mix ing me c ha n i sm e xp l a ine dthe a n i so t r opy o f a num be r o f N i ba sed a l l oys .

We can summar ize the conc lus ions we can drawfrom the ana lys is g iven here .(1) For impur i t ies for which there a re no d s ta tes

a t the spin 1 ' Fermi sur face (non- t rans i t ion impur i t ies ,Co , Fe , M n, Pd) , t he Sm i t me c ha n i sm a lone p r ov ide sa good d e sc r ip t i on o f t he a n i so t r opy w i th A p ~ =7( ~ - 1 ) , w he r e 3' is a sp in - o r b i t pa r a me te r w hic h w ef ind e qua l t o a bou t 0 .01 0 . A l t e r na ti ve ly , i f s e pa r a t eva lues of ( A p / p ) t a nd ( A p / p ) ~ have been obta inedf rom an ana lys is of te rnary a l loy da ta , then for theseimpur i t i e s( A p / p ) t ~ 7 a , ( A p /p ) ~ " . - 7 ,so t ha t A p~ = - - A pt .

(2) For other t r an s i t ion impu r i t ies wh ere there i s aVBS at the spin 1' Fermi surface, i t is necessary to in-t r oduc e a n a dd i t iona l a n i so t r opy i n the sp in 1' ba ndres is t iv i ty , so tha t we wr i te ( A p / p ) t = 7or + 3/3. llere,a s de t e rmine d e xpe r ime n ta l l y /3 var ie s f rom imp ur i t yto impur i ty and can be e i ther s ign.

We have suggested t ha t the 13 te rm can be ident i f iedw i th a n L z S z s p i n - o r b i t m e c h a n is m a c t in g w i t h i n th e


11/12

O . J a o u l e t a i. / S p o n t a n c o u s r e s i s t i v i t y a n i s o t r o p y i n N i a l l o y s 3 3spin t V B S . W e h a v e d e v e l o p ed a s i m p l e m o d e l w h i c hg ive s the c o r r e c t o r de r o f m a gn i tude o f t he e f f e c t , t o -g e t h e r w i t h a n i n d i c a t io n o f h o w t h e e f f e c t c a n b e ex -p e c t e d t o v a r y f r o m i m p u r i t y t o i m p u r i t y a c r o s s at r a ns i t i on se r i e s .J u d g i n g f r o m t h e e x p e r i m e n t a l r e s u l ts , t h e re a p -pe a r s t o be no ne e d to in t r oduc e a n a na logous sp in 4 ,t e r m , so tha t f o r t he se im pur i t i e s a l so ( A p / p ) ~ , ~ - " 7 .

F r o t h s y m m e t r y a r g u m e n t s , i f t h e r e s is t iv i t y an -i s o t ro p y i s d u e t o a w e a k s p i ~ -- o r b i t p e r t u r b a t i o n o ft h e s t r u c t u r e o f t h e m e t a l , th e l e a d i n g t e r m s m u s t b es e c o n d - o r d er i n s p i n o r b i t , t h a t i s t o s a y o f t h e ty p e( M , + S _ ) 2 o r ( M , r S z ) 2 . The f o r m e r i s t he S m i t m e -c h a n i sm t e r m , a n d t h e l at t e r t h e a d d i t i o n a l a n i s o t r o p yin a sp in 1' VB S . E ve n thou gh w e ha ve use d a ve r ys im ple m ode l o f t he e l e c t r on ic s t r u c tu r e o f ~ he a l loys ,t h e i d e n t i f i c a ti o n o f t h e t y p e o f m e c h a n i s m l e ad i n g t othe a n i so t r opy in va rious c a se s se e m s c l e a r , A f u r the ra na lys is on s ing le c r ys t a l s shou ld g ive m or e de ta i l e di n f o r m a t i o n a b o u t t h e se m e c h a n i s m s i n r e l a ti o n t o ar e a l i s t i c ba nd s t r uc tu r e sc he m e inc lud ing sp in o r b i t .

A c k n o w l e d g e m e n t sWe wo uld l i ke to tha nk P r o f e s so r P e r io a nd Dr s .

La ur i a t a nd Dixm ie r f o r k ind ly gu id ing us in the t e x -t u re p r o b l e m .

A p p e n d i xS upp ose w e ha ve a se t o f t e r na r y Ni ba sed a l loys

N i t _ x . y A x B y , w h e re a A = ( O o ~ / P o t ) ~ a n d % ={ p o U P o t ) , a re k n o w n.

We wi ll a s sum e tha t a l l t he a l loys a r e su f f i c i e n t lyc onc e n t r a t e d f o r t he e f f e c t s d i sc usse d in se c t ion 4 tobe un im por t a n t . T i l e r e s i s t i v i ty a n i so t r op ie s on the sea l loys c a n be p lo t t e d a ga ins t t he r e l a t ive c onc e n t r a -

l. ^ = CA/i ons o f B to A , e xpr e s se d by t ,, ,~ pa . . . . . . ' -dt I I I~., I ~., I t( C a + C u ) whic h va r i e s f r om 0 to I as x / y varies from0 to oo. We define.A=(aolp) , d A = ( A p / p ) ' ~ ~'u B = ( x p /o )P , d n = ( A P / P ) ~ T o d e t e r m i n e t h e s e f o u r p a r a m e t e r s s e p a r a te l y , w e

~ ~ 9 / ~ ( '/ . )8 L

t -

6 ;

; toE0 112

. . , . . . _ - . . - . . . . . . - - .

A ~ ,,t __0.4 0.6 0.8 tI "i g. A I . T h e r e s i s t iv i t y a n i s o t r o p y o f N i t _ . , : v M n x C r v a l l o y sa t 4 . 2 K a s a f u n c t i o n o f t h e r e la t iv e c o n c c n l t a t i o n t = x x + rT h e c u r ve is b a s e d o n d a t a b y D o r l ei j n a n d \ l i e d c m a 1 81 a n dJ a o u l [ 1 5 1.c a n u s e t h e f o u r e x p e r i m e n t a l p a r a m e t e r s( A P / o ) A ' ( A P / p ) B " ~ W ] : - - . O -d-t~ p l , - t( se e f ig . A I ) . I t c a n be show n by a l i t t l e a lge b r a tha tw e h a v e t h e f o u r s i m u l t a n e o u s e q u a t i o n s :

( i ) u A + ( l / a A ) d A = (o~A + i /0 t a ) ( A p / p ) A .( i i ) uu + (I /oq~) dB = (o~u + l /a te ) ( A P / P ) B ,

( i i i ) U A [ ~ 3 ( q ~ + 1 ) l + u l ~ [ a A + 1 - a u ( a j 3 + 1 ) 2 ]

+ dA (a B + I ) ~A dB(CtA(20~B+ 1 )= ( a a + l ) 3 ( ~ A P ~ d A p )

d t , - o

( i v ) U A [0 ~ A ( a A + [ ) 2 _ ( C t B + 1 ) 1 - UBIe2(C~A 1) !- d A [ 0 t 2 - - ( 2 0 t A + l )O t B ] - d B ( O t B ( e A + 1 ))

u [ , , ~ ,- l&tA

' ' ' ~ a a + 1 pO d t p l t _ _ ,wh ere O~,, O a re the spec i f ic res is t iv i t ies per p ercentim pur i ty f o r im pur i t i e s A , B . The se f ou r e qua t ionsc a n b e s o lv e d i m m e d i a t e l y f o r t h e f o u r u n k n o w n s , i ti s i n s t r uc t ive to pu t i n the num e r i c a l f a c to r s f o r t hecase show n in ig . A l , w here A is Mn, B is Cr . Le t ust a ke a s g iven f o r i n s t a n c e a M n = 11 , a c t = 0 . 35 . Th i s


12/12

34 O . . l a o u l cta l , / Spontatu 'ous re sis t ivi ty anisotrop y in ,Vi alko ,si s a t y p i c a l c a s e w l t e r e e A ~ ' 1, oq~ < I . W e h a v e

( i ) u . x + O. Oq ' I A = ! . ! ( A p / p ) A = 10 . 5 ,

( i i ) t t B + ' . 8 5 d B = 3 . 9 ( A p / p ) u . . . . 0 . 8 ,{i ii~ { 10)UA + t ! I .35)UII + ( ! 4 . 9) d A - - ( 1 8 .6 )d B

( . . _ 6 1 1 6 . 8 t d A p --*o

d A = OtA/(C~A - - 0 tB) 3 [ {a A + aB ) K0 - 20 lB KI ] ,

d B = 0tB/(OtA -- OrB)3 ( 2 a A K o - (C~A + OR B ) K I ) ,w h e r e U A a n d u u c a n t h e n b e d e t e r m i n e d f r o m e q s . ( i )a n d ( i i ) .

A p p l i c a t i o n o f e i t h e r o f th e s e m e t h o d s s h o w s t h a tt h e a c c u r a c y o f t h e f i n a l e s t i m a t e s r e q u i r e s p o i n t s v e r yc l o s e t o t h e t = i s i d e o f t h e f i g . A I .

( iv ~, 1 5 t ) 0 u a 1 4 5 0 u B - l i 3 d A - - 4 d B

= - 6 7 s - ~ ' - a i - 7 - - - - !

I t c a n b e s e e n t h a t s o l v i n g e q u a t i o n s ( i ) , ( i v ) , ( i i ) a n dt i i i ) s u c c e s s i v e l y g i v e s e s s e n t i a l l y t h e p a r a m e t e r s U A ,u B, d B and ,t:x i n t h a t o r d e r . The e r r o r s a r i s e ma i n l yf r o m z h e u n d e r t a i n t i e s i n t h e e s t i m a t e s o f t h e l i m i t i n gs l o p e s . I t e r e w e f i n du A = 1 0 .5 " U B = ! . 2 " d A = - 1 . 6 ; d B = - 0 . 7 .A n a l t e rn a t i v e b u t e q u i v a l e n t m e t h o d w h i c h h a s t h ea d v a n t a g e o f t ,s i n g a ll t h e m e a s u r e d p o i n t s i s t h e f o l-l o w i n g . D e f i n e k = t t p tB) do tha t o~P A / ( P A + . = ' ~ ' 0 : A +( 1 -- M e u. a n d X g o e s f r o m 0 t o 1 as t g o e s f r o m 0 to 1.T h e n-..&p _o ~ + [ ' ~ : A + ( l - X ) U B + Xa Ad A + (1 - - ;k ) a B d B0:2

T h i s c a n b e r e w r i t t e ne: : a + 1 Apx ( 1 - - ~ ) - - - p

aA + I ( ~ Q 1 1 - - X ) a B + 1 ( ~

= CtB{-' -,~ ) 2}, CtB (1 X ) ( C t 2 / ~ A ) d A

+ aB ( 2 h) 2 ( 1 ~ ) 0 :A ~k( O~2/O~B)dB T!~ ~ e x p c d m e n t a l ! y d e t e r m i n e d f u n c t i o n o n t h e l e f t -h a n d s id e is l i n e a rl y d e p e n d e n t o n X , w i t h i n t e r c e p t sK o w h e n k = O , K l w h e n ? ~ = ! . F ' r o m t h e s e i n t e r c e p t s

R e f e r e n c e s[ 1 ] I .A. Cam pbell, A . l:ert and A.R . Pom eroy, P hil. Ma g. 15

( 1967) 977 .121 P. Leonard , M.C. Cadeville, J . D urand and I:. Ga utie r ,J . Phys . Chem. Solids 30 (196 9) 2169.13 I . ' .C. Schwerer and J. Co nro y, J. Phys. !"1 (197 1) 877 .[4 J.I. '.W. Dorleijn and A.R . M iedema , J. Phys. i"5 { 1975 )4 8 7 .[5 A. l. 'er t and i.A. Cam pbell, J . I 'hys. 1'6 (197 6) 849 .[61 J. Smit, Physica 16 (1951) 612.[7 i.A. Campbell, A. I :ert and O. Jaoul, J . Phys. C3 (Met.Phys. Suppl.) 119.70) $95.[8 J.I .W. Dorleijn and A.R. Micdema, J. I 'hys. 15 (197 5)1543.19 I".C. Schwcrer and .I . Silcox , l 'hys. Rcv. BI (197 0) 2391 .101 J. i. . Snoek, Nature 163 (1949) 411.! !1 il.C . Van Elst, I 'hysica 25 (19 59 ) 708.121 Y. Shirakawa, Sci . Rep. Toho ku Univ. 27 (1939) 485.131 I".R. Mc (;uire, A ll ' Con f. Proc. 24 (1975 ) 43 5.141 L. Berge~ and S.A. i:r ied berg, Pl~ys. Rev . 165 (1 968 )6 7 0 .I151 O. Ja oul , Thes is , Orsay (19 74) .1161 P.L. Maitrepicrre, J . Appl. l 'hys. 40 (1969) 4826.! 17] O. Jao ul and I .A. Cam pbell, J . Phys. 1'5 (197 5) L 69.I 18 ] ! :. 13rouers, A.V . Ve Jya ver and M. ( ;iorgino, Phys. Rcv.B 7 (1973) 380.

I i 91 T. I .arrell and D. ( ;reig, J. Phys. C 1 (1968) 1359.1201 A. I :ert and I .A. Cam pbell, P hys. Rc v. Lctt. 21 (1968 )1 1 0 9 .1211 (;.( ;. I 'L Low, J. Appl. Phys. 36 (1965) 1174.[221 J. l"riedei, Nuov o CmL 7, Suppl. 2 (1958) 287.1231 J. Kanamori, J . Appl. Pl ' ,ys. 36 (1965) 929.

. . . . . '~. t t t tl t , u~ a tl i_ ll lt l t~ t . l '~ . ( J t ) l l i C S , P r o c . Pilys. Soc. 9 i(1967) 319.[251 A.A . ( ;omcs and 1. ~,. Cam pbe ll, J . I 'hys. ( '3 (196 8) 1312.1261 M .C. Cadeville, i: . ( ;au tier , C. R obe rt and J. R oussel,Solid State Comnmn. 7 (1969) 1701.1271 R. Caudron, R. Caplain, J.J. Meunier and P. Costa, Phys.Rev. 68 (1973) 5247.1281 A. I:riederich and A . l.'ert, Phys. R ev . Lctt. 3"1 (19 74 )1214 .[:]91 G.A . ( ;ehring, Phys. Scr. I i (1 975 ).1:301 J . Du rand, Thesis , S trasbourg (1973 ) .

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