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Anomalies due to anisotropic s-d exchange interaction
Volume 22, number 4
ANOMALIES
PHYSICS
DUE
TO
LETTERS
ANISOTROPIC
1September1966
s-d E X C H A N G E
INTERACTION
H. M I W A
Department of Physics, Osaka University, Toyonaka, Osaka, Japan and Y. NAGAOKA
Department of Physics, Nagoya University, Nagoya, Japan Received 26 July 1966
Significance of studying the anisotropic s-d exchange interaction is pointed out. The most divergent logarithmic terms are summed up. Additional divergence difficulty is suggested for the case of ferromagnetic coupling.
Since the a n o m a l o u s t e m p e r a t u r e dependence of the r e s i s t i v i t y of dilute alloys with m a g n e t i c i m p u r i t i e s was explained by Kondo [1], the l o g a r i t h m i c d i v e r g e n c e s of the p e r t u r b a t i o n e x p a n s i o n in v a r i o u s q u a n t i t i e s have been i n v e s t i g a t e d by m a n y a u t h o r s [2-9]. In this note we point out the i m p o r t a n c e of studying the effect for the case of an a n i s o t r o p i c exchange i n t e r a c t i o n : Hsd = -(1/N)~k ,k' S. J. a*le,crak,
(1)
where S i s the spin o p e r a t o r for a localized spin, a k and a k a r e , r e s p e c t i v e l y , the a n n i h i l a t i o n (column) and the c r e a t i o n (row) v e c t o r s for the conduction e l e c t r o n state k with two spin c o m p o n e n t s , and a is the P a u l i m a t r i x . F i n a l l y , J i s the exchange i n t e r a c t i o n t e n s o r which i s a s s u m e d to be independent of k. V e r y r e c e n t l y , Sugawara et al. [10] o b s e r v e d log T - t e r m s in e l e c t r i c a l r e s i s t i v i t y of m a n y dilute a l l o y s of y t t r i u m and l a n t h a n u m with m a g n e t i c r a r e - e a r t h e l e m e n t s . As the c r y s t a l l i n e field splitting of the 4 / l e v e l s i s l a r g e r or c o m p a r a b l e with the t e m p e r a t u r e where the log T t e r m s a r e i m p o r t a n t , it is e s s e n t i a l for a n a l y s i s of the data to take the splitting into account. If the t e m p e r a t u r e i s sufficiently low, we can l i m i t o u r s e l v e s within the ground m u l t i p l e t of the 4/ l e v e l s . Then, in t e r m s of an effective g - t e n s o r for an effective spin S of the ground m u l t i p l e t , the tensor J is written as J = Jsf(1 - 1/g)geff,
Veff~ -[~(E)Sxa x + ~2 (~)SyO'y +Jz(E)Sz(~z ],
;
:
(3)
Jy'
<4>
\jyj~o / j where t = (p/N)log(IE I / ~ F ) and we have chosen a coordinate s y s t e m where J b e c o m e s a diagonal t e n s o r . An explicit e x p r e s s i o n for ~x, for example i s given by y
=
x
z
(5)
From eqs. (3) and (5), we obtain an expression for the r e s i s t i v i t y due to m a g n e t i c i m p u r i t i e s as
PM = LPo/ o )tv~ x + y
log ~ +
(2)
where Jsf i s the i s o t r o p i c exchange constant for
394
a f r e e 4/ spin and the t e n s o r geff i s g e n e r a l l y a n i s o t r o p i c for the r a r e - e a r t h s p i n s in a h e x a gonal or a t r i g o n a l e n v i r o n m e n t . It is i n t e r e s t i n g to see how A b r i k o s o v ' s r e sult i s modified for this case when the m o s t d i v e r g e n t t e r m s for the s m a l l conduction e l e c t r o n e n e r g y in each o r d e r of J a r e s u m m e d up. F o l lowing Doniach [7], we get the effective i n t e r a c tion which i n c l u d e s all the m o s t d i v e r g e n t t e r m s :
+
+
+
~og~F /
..-],
(6)
Volume 22, number 4
P HY SI C S L E T T E R S
where Po i s the r e s i s t i v i t y c a l c u l a t e d in the Born a p p r o x i m a t i o n for i s o t r o p i c exchange i n t e r a c t i o n Jo" The c o e f f i c i e n t s can be ex~oressed in i n v a r i a n t forms Sp(j2), det J l and [Sp(J~)]2_ Sp(j4), respectively. It should be noticed here that the exchange constants appear in each order in different combinations. The resitance m i n i m u m is expected if det]Jl< 0. F r o m the theoretical point of view, Vef f has s o m e interesting properties. If the denominator of eq. (5) vanishes for s o m e negative values of t, we m e e t the s a m e difficult situation a s that for the i s o t r o p i c a n t i f e r r o m a g n e t i c c a s e , that is, a s o r t of r e s o n a n c e s or a c e r t a i n kind of bound s t a t e s [2,3,4,8,9] whose e s s e n t i a l n a t u r e i s not completely understood. Some s p e c i a l c a s e s should be noted: 1) F o r J x = J y = J z = J (isotropic case), eq. (5) r e d u c e s to a well-known f o r m = J/(i
- 2tJ).
(7)
2) F o r J x = Jy = 0 (Ising case), we get ~ = J and all d i v e r g e n t t e r m s v a n i s h a s expected. 3) F o r J z = O, J x ¢ O, J y ¢ 0, we obtain JX[1 + t 2 ( J 2 - Jx2)] ,
:
1 September 1966
(which m a y be c a l l e d a s f e r r o m a g n e t i c case) on the negative t - a x i s . H e r e , a p a r t i c u l a r attention should be paid to the " a l m o s t f e r r o m a g n e t i c " case, that i s , to J deviating slightly an i s o t r o p i c J(>0). In this case we have two p o l e s together at t ~ - I / J , but the r e s i d u e s for these poles v a n i s h when J t e n d s to be i s o t r o p i c . Thus, we may r e g a r d the i s o t r o p i c f e r r o m a g n e t i c case a s to have a l a t e n t pole twice a s f a r a s the r a d i u s of c o n v e r g e n c e for the o r i g i n a l s e r i e s . B e s i d e s the question on the c o n v e r g e n c e of the a l t e r n a t i n g s e r i e s , p h y s i c a l m e a n i n g s , if any, of this pole which n a r r o w l y e s c a p e d f r o m p r e v i o u s i n v e s t i g a t i o n s should be made c l e a r in the c o n c l u s i v e theory a s well a s that for the c o n t r o v e r sial a n t i f e r r o m a g n e t i c case. T h i s work was p e r f o r m e d when the a u t h o r s were staying at the Institute for Solid State P h y s i c s , U n i v e r s i t y of Tokyo. They would like to e x p r e s s t h e i r thanks to P r o f e s s o r K. Yosida, P r o f e s s o r A . Y o s h i m o r i and M r . A. Okiji for t h e i r hospitality and for v a l u a b l e d i s c u s s i o n s . They a r e a l s o indebted to P r o f e s s o r T. Sugawara and Dr. H. Nagasawa for d i s c u s s i o n s on t h e i r e x p e r i m e n t a l data.
(s)
4) F o r J x = J y = J J- ~ Jz = JII (axially s y m m e t r i c case), we obtain J,= 1 - f.~
J.(i
- 2t2j~l'~
+ ~l )
1 - tJL~ - 2t2j'~ ~ T h u s , for c a s e s 3) and 4), we always have a n e g a tive v a l u e of t where the d e n o m i n a t o r v a n i s h e s . The c l o s e e x a m i n a t i o n of eq. (5) for the g e n e r a l case other than the above four shows that t h e r e i s one pole for d e t ] J ] < 0 (which may be called a s a n t i f e r r o m a g n e t i c case) and two p o l e s for det J >0
References 1. J. Kondo, Prog. Theor. Phys. Kyoto 32 (1964) 37. 2. H.Suhl, Phys. Rev. 138 (1965) A515; Physics 2 (1965) 39; Phys. Rev. 141 (1966) 483, and to be published. 3. Y.Nagaoka, Phys. Rev. 138 (1965)All12. 4. A.A.Abrikosov, Physics 2 (1965) 5. 5. K.Yosida and A.Okiji, Prog. Theor. Phys. (Kyoto) 34 (1965) 505. 6. K. Ishikawa and Y.Mizuno, Prog. Theor. Phys. (Kyoto) 35 (1966) 746. 7. S.Doniach, Phys. Rev. 144 (1966) 382; Bull. Am. Phys. Soc. Ser. II, 11 (1966) 387. 8. K.Yosida, Phys. Rev. to be published. A.Okiji, Prog. Theor. Phys. (Kyoto)to be published. 9. J. Kondo, private communication. 10. T. Sugawara, J. Phys. Soc. Japan 20 (1965)2252; T. Sugawara and H.Eguchi, J. Phys. Soc. Japan 21 (1966) 725.
*****
395
ANOMALIES
PHYSICS
DUE
TO
LETTERS
ANISOTROPIC
1September1966
s-d E X C H A N G E
INTERACTION
H. M I W A
Department of Physics, Osaka University, Toyonaka, Osaka, Japan and Y. NAGAOKA
Department of Physics, Nagoya University, Nagoya, Japan Received 26 July 1966
Significance of studying the anisotropic s-d exchange interaction is pointed out. The most divergent logarithmic terms are summed up. Additional divergence difficulty is suggested for the case of ferromagnetic coupling.
Since the a n o m a l o u s t e m p e r a t u r e dependence of the r e s i s t i v i t y of dilute alloys with m a g n e t i c i m p u r i t i e s was explained by Kondo [1], the l o g a r i t h m i c d i v e r g e n c e s of the p e r t u r b a t i o n e x p a n s i o n in v a r i o u s q u a n t i t i e s have been i n v e s t i g a t e d by m a n y a u t h o r s [2-9]. In this note we point out the i m p o r t a n c e of studying the effect for the case of an a n i s o t r o p i c exchange i n t e r a c t i o n : Hsd = -(1/N)~k ,k' S. J. a*le,crak,
(1)
where S i s the spin o p e r a t o r for a localized spin, a k and a k a r e , r e s p e c t i v e l y , the a n n i h i l a t i o n (column) and the c r e a t i o n (row) v e c t o r s for the conduction e l e c t r o n state k with two spin c o m p o n e n t s , and a is the P a u l i m a t r i x . F i n a l l y , J i s the exchange i n t e r a c t i o n t e n s o r which i s a s s u m e d to be independent of k. V e r y r e c e n t l y , Sugawara et al. [10] o b s e r v e d log T - t e r m s in e l e c t r i c a l r e s i s t i v i t y of m a n y dilute a l l o y s of y t t r i u m and l a n t h a n u m with m a g n e t i c r a r e - e a r t h e l e m e n t s . As the c r y s t a l l i n e field splitting of the 4 / l e v e l s i s l a r g e r or c o m p a r a b l e with the t e m p e r a t u r e where the log T t e r m s a r e i m p o r t a n t , it is e s s e n t i a l for a n a l y s i s of the data to take the splitting into account. If the t e m p e r a t u r e i s sufficiently low, we can l i m i t o u r s e l v e s within the ground m u l t i p l e t of the 4/ l e v e l s . Then, in t e r m s of an effective g - t e n s o r for an effective spin S of the ground m u l t i p l e t , the tensor J is written as J = Jsf(1 - 1/g)geff,
Veff~ -[~(E)Sxa x + ~2 (~)SyO'y +Jz(E)Sz(~z ],
;
:
(3)
Jy'
<4>
\jyj~o / j where t = (p/N)log(IE I / ~ F ) and we have chosen a coordinate s y s t e m where J b e c o m e s a diagonal t e n s o r . An explicit e x p r e s s i o n for ~x, for example i s given by y
=
x
z
(5)
From eqs. (3) and (5), we obtain an expression for the r e s i s t i v i t y due to m a g n e t i c i m p u r i t i e s as
PM = LPo/ o )tv~ x + y
log ~ +
(2)
where Jsf i s the i s o t r o p i c exchange constant for
394
a f r e e 4/ spin and the t e n s o r geff i s g e n e r a l l y a n i s o t r o p i c for the r a r e - e a r t h s p i n s in a h e x a gonal or a t r i g o n a l e n v i r o n m e n t . It is i n t e r e s t i n g to see how A b r i k o s o v ' s r e sult i s modified for this case when the m o s t d i v e r g e n t t e r m s for the s m a l l conduction e l e c t r o n e n e r g y in each o r d e r of J a r e s u m m e d up. F o l lowing Doniach [7], we get the effective i n t e r a c tion which i n c l u d e s all the m o s t d i v e r g e n t t e r m s :
+
+
+
~og~F /
..-],
(6)
Volume 22, number 4
P HY SI C S L E T T E R S
where Po i s the r e s i s t i v i t y c a l c u l a t e d in the Born a p p r o x i m a t i o n for i s o t r o p i c exchange i n t e r a c t i o n Jo" The c o e f f i c i e n t s can be ex~oressed in i n v a r i a n t forms Sp(j2), det J l and [Sp(J~)]2_ Sp(j4), respectively. It should be noticed here that the exchange constants appear in each order in different combinations. The resitance m i n i m u m is expected if det]Jl< 0. F r o m the theoretical point of view, Vef f has s o m e interesting properties. If the denominator of eq. (5) vanishes for s o m e negative values of t, we m e e t the s a m e difficult situation a s that for the i s o t r o p i c a n t i f e r r o m a g n e t i c c a s e , that is, a s o r t of r e s o n a n c e s or a c e r t a i n kind of bound s t a t e s [2,3,4,8,9] whose e s s e n t i a l n a t u r e i s not completely understood. Some s p e c i a l c a s e s should be noted: 1) F o r J x = J y = J z = J (isotropic case), eq. (5) r e d u c e s to a well-known f o r m = J/(i
- 2tJ).
(7)
2) F o r J x = Jy = 0 (Ising case), we get ~ = J and all d i v e r g e n t t e r m s v a n i s h a s expected. 3) F o r J z = O, J x ¢ O, J y ¢ 0, we obtain JX[1 + t 2 ( J 2 - Jx2)] ,
:
1 September 1966
(which m a y be c a l l e d a s f e r r o m a g n e t i c case) on the negative t - a x i s . H e r e , a p a r t i c u l a r attention should be paid to the " a l m o s t f e r r o m a g n e t i c " case, that i s , to J deviating slightly an i s o t r o p i c J(>0). In this case we have two p o l e s together at t ~ - I / J , but the r e s i d u e s for these poles v a n i s h when J t e n d s to be i s o t r o p i c . Thus, we may r e g a r d the i s o t r o p i c f e r r o m a g n e t i c case a s to have a l a t e n t pole twice a s f a r a s the r a d i u s of c o n v e r g e n c e for the o r i g i n a l s e r i e s . B e s i d e s the question on the c o n v e r g e n c e of the a l t e r n a t i n g s e r i e s , p h y s i c a l m e a n i n g s , if any, of this pole which n a r r o w l y e s c a p e d f r o m p r e v i o u s i n v e s t i g a t i o n s should be made c l e a r in the c o n c l u s i v e theory a s well a s that for the c o n t r o v e r sial a n t i f e r r o m a g n e t i c case. T h i s work was p e r f o r m e d when the a u t h o r s were staying at the Institute for Solid State P h y s i c s , U n i v e r s i t y of Tokyo. They would like to e x p r e s s t h e i r thanks to P r o f e s s o r K. Yosida, P r o f e s s o r A . Y o s h i m o r i and M r . A. Okiji for t h e i r hospitality and for v a l u a b l e d i s c u s s i o n s . They a r e a l s o indebted to P r o f e s s o r T. Sugawara and Dr. H. Nagasawa for d i s c u s s i o n s on t h e i r e x p e r i m e n t a l data.
(s)
4) F o r J x = J y = J J- ~ Jz = JII (axially s y m m e t r i c case), we obtain J,= 1 - f.~
J.(i
- 2t2j~l'~
+ ~l )
1 - tJL~ - 2t2j'~ ~ T h u s , for c a s e s 3) and 4), we always have a n e g a tive v a l u e of t where the d e n o m i n a t o r v a n i s h e s . The c l o s e e x a m i n a t i o n of eq. (5) for the g e n e r a l case other than the above four shows that t h e r e i s one pole for d e t ] J ] < 0 (which may be called a s a n t i f e r r o m a g n e t i c case) and two p o l e s for det J >0
References 1. J. Kondo, Prog. Theor. Phys. Kyoto 32 (1964) 37. 2. H.Suhl, Phys. Rev. 138 (1965) A515; Physics 2 (1965) 39; Phys. Rev. 141 (1966) 483, and to be published. 3. Y.Nagaoka, Phys. Rev. 138 (1965)All12. 4. A.A.Abrikosov, Physics 2 (1965) 5. 5. K.Yosida and A.Okiji, Prog. Theor. Phys. (Kyoto) 34 (1965) 505. 6. K. Ishikawa and Y.Mizuno, Prog. Theor. Phys. (Kyoto) 35 (1966) 746. 7. S.Doniach, Phys. Rev. 144 (1966) 382; Bull. Am. Phys. Soc. Ser. II, 11 (1966) 387. 8. K.Yosida, Phys. Rev. to be published. A.Okiji, Prog. Theor. Phys. (Kyoto)to be published. 9. J. Kondo, private communication. 10. T. Sugawara, J. Phys. Soc. Japan 20 (1965)2252; T. Sugawara and H.Eguchi, J. Phys. Soc. Japan 21 (1966) 725.
*****
395