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Effect of electron correlations on interaction of localized magnetic moments
Volume 30A, n u m b e r 8
PHYSICS
ion distribution over the magnetic sublattices. T h e d i f f e r e n c e i n e l e c t r o n i c s h e l l s t r u c t u r e of m e t a l i o n s of t h e s e s u b s t a n c e s ( g r o u n d s t a t e s 6S_~ a n d 4F{ ) s e e m s t o b e i m p o r t a n t too. W h i l e t h e 2 d e p e n d e n c e of F e 2 0 3 m a y b e a c c o u n t e d f o r b y t h e m a g n e t o - d i p o l e i n t e r a c t i o n [5], t h e c o r r e s p o n d i n g c a l c u l a t i o n s f o r C r 2 0 3 p r e d i c t a v a l u e [2] w h i c h i s b y a n o r d e r of m a g n i t u d e l e s s t h a n t h e o b s e r v ed one. It c a n b e s u p p o s e d t h a t t h e e f f e c t o b s e r v e d m a y be accounted for by the anisotropic energy contribution connected with the spin-orbit interaction i n t h e c r y s t a l . T h e p r e s e n t t h e o r y i s u n a b l e to produce a quantitative estimate. However, one may expect that in a crystal with a large supere x c h a n g e i n t e r a c t i o n (T N = 3 0 8 ° K ) t h e p r o p e r t i e s of i t s a n i s o t r o p i c p a r t [6] a r e s i g n i f i c a n t in p a r -
EFFECT
OF ELECTRON OF LOCALIZED
LETTERS
15 December 1969
ticular. They are noticeable against the small v a l u e of t o t a l a n i s o t r o p y e n e r g y . We a r e g r a t e f u l t o V. M. F r i d m a n f o r t h e assistance in the experiment. We have benefited f r o m d i s c u s s i o n s w i t h A. S. B o r o v i k - R o m a n o v .
References 1. s. F o n e r and S. L. Hou. J Appl. Phys• 33 (1962) 1289S. 2. J O. Artman. J. C Murphy and S. Foner. Phys. Rev. 138 {1965) 912• 3. J. S. J a c c b s and P. E. Lawrence. Rev. Sei. Instr• 29 (1958) 713. 4. T.G. Worlton. R. M. B r u g g e r and R. B. Bennion. J. Phys• Chem. Sol. 29 (1968) 435. 5. P.A. Voskanian. R. Z. Lewitin and V A Shchurov, Zh. Eksp. i. Teor. Fiz. 54 {1968) 790. 6. T. Moriya, Phys. Rev. 120 {1960) 91.
CORRELATIONS MAGNETIC
ON INTERACTION MOMENTS
J. M O R K O W S K I a n d A. S T A S C H Ferroraagnetics Laborator3.,, Institute of Fhysics of the Polish Academy of Sciences, Pozna~, Poland Received 3 November 1969
R is shown that the effect of e l e c t r o n c o r r e l a t i o n s is to change the c r i t e r i o n for o c c u r r e n c e of localized m a g n e t i c states. The conditions of existence a r e different for p a r a l l e l and a n t i p a r a l l e l configurations of a p a i r of localized magnetic states. In g e n e r a l , the p a r a l l e l configuration is favoured•
W e c o n s i d e r t h e A n d e r s o n [1,2] m o d e l of two impurity transition metal atoms, each having a single, non-degenerate d-state, introduced into a m a t r i x of a n o n - m a g n e t i c m e t a l . T h e s y s t e m of c o n d u c t i o n a n d d - e l e c t r o n s i s d e t e r m i n e d b y t h e t i a m i l t o n i a n ( t h e n o t a t i o n of r e f . 2 i s u s e d throughout).
~g=k~aCknko. +~. E o n i a $a
+ (~ Vikq + ika Cka. C~a r e f e r Ct*a c o r r e s p o n d n -ka - = C k+a C ka ,
+
_t
~. 2Vnicrni,-~r + ,a
(1)
Ck~ + ~ v 1 2 C ~ C 2 ~ + c . c . ) .
t o t h e c o n d u c t i o n e l e c t r o n s , Ci , to the localized d-electrons an~ n -,~ = C ,-+~ , a•" Ek i s t h e e n e r g y of c o n d u c t i o n e l e c t r o n s , E o d e n o t e s t h e e n e r g y of the localized d-state, U is the Coulomb interact i o n e n e r g y ; Vik a n d V12 a r e t h e m a t r i x e l e m e n t s for t h e m i x i n g i n t e r a c t i o n of e l e c t r o n s . W e assume V >> I V/k l, IV121-
46O
T h e e x p e c t a t i o n v a l u e s (n/a~" a r e c a l c u l a t e d by t h e G r e e n f u n c t i o n m e t h o d [4,3]• T h e c h a i n of t h e e q u a t i o n s of m o t i o n f o r t h e G r e e n f u n c t i o n s (( Cicr; C~)) i s r e d u c e d to a c l o s e d s y s t e m b y • . + n e g l e c t i n g t h e G r e e n f u n c t i o n s ((Ck -a Ci~ Ci ~; 4: + + + 7C~.~) a n d (( C~ _~ C~.~ C, _~,; C , ~ ) (ct. [3]), a n d b y th~e~ f o l l o w i n g ~ d e ~ o u ~ l i @ s Y o-
+ => + F o r U >> IV/k I,
forj
¢ i.
IV12 ] we f i n d f i n a l l y
(nia) _ 1 - (ni,_~} x n Vrctan ×
La
(2)
EF - E 0 - ( 1 -) (X- IV12 125 A(1
-
Volume 30A, n u m b e r 8
PHYSICS
i
. .o . . , . ,
f2 r;g 1o
f3
-Of
N
Fi~ ~b
I
I
I
1
2
3
I
X: A Fig. la. Regions of existence of magnetic and n o n m a g netic solutions of eq. (2). The dotted c u r v e c o r r e s p o n d s to n o n - i n t e r a c t i n g i m p u r i t i e s (z = 0), the two other a r e calculated for z = 0.1. In the region above the full line t h e r e a p p e a r localized magnetic m o m e n t s coupled p a r allel; s i m i l a r l y the b r o k e n line d e t e r m i n e s the domain of existence for a n t i p a r a l l e l configuration of localized magnetic m o m e n t s . Fig. lb. Relative stability of the p a r a l l e l and a n t i p a r allel configurations of localized magnetic m o m e n t s . The two c u r v e s of the e n e r g y difference E . - E , v e r s u s the F e r m i energy or the p a r a m e t e r x w e r e calculated for y = 2.25 and 3.0, respectively. H e r e ( i , j ) = ( 1 , 2 ) o r (2, 1); E F d e n o t e s t h e F e r m i ener~, a n d ~ a n d ,x a r e d e f i n e d b y ~ - iA = = ~ k [ 1 V/k t 2 / ( E F - Ek + i0+)] • (We a s s u m e t h a t a n d ,x a r e s l o w l y v a r y i n g f u n c t i o n s of e n e r g y a n d we t a k e t h e i r v a l u e s f o r E = E F ) . F o l l o w i n g r e f . 2 we s e e k f o r t h e s e l f - c o n s i s t e n t s o l u t i o n s
LETTERS
15 D e c e m b e r 1969
( n l ~ } = ( n 2 a ) = n c r a n d ( n l a > = ( n 2 , _ ~ = m a of eq. (2~, c o r r e s p o n d i n g to p a r a l l e l a n d a n t i p a r a l l e l c o u p l i n g of t h e two l o c a l i z e d m o m e n t s , r e s p e c t i v e l y . T h e m a g n e t i c s o l u t i o n s h a v e to s a t i s f y t h e c o n d i t i o n s n~ - n; ¢ 0 o r m~ - m; ¢ 0. T h e r e s u l t s of t h e t h e o r y a r e d i s c u s s e d in t e r m s of t h e f o l l o w i n g d i m e n s i o n l e s s p a r a m e t e r s : x = ( E o - E F ) / A , y = -~,/A, z = IV12 12,yUA. F r o m r e f . 2 t h e r e g i o n s of t h e s e p a r a m e t e r s c a n b e found, for which there appear localized magnetic states coupled paralle] or antiparallel. There are regions for which both configurations are adm i s s i b l e . In s u c h c a s e s it i s n e c e s s a r y to c a l c u l a t e t h e e n e r g y of t h e s e two c o n f i g u r a t i o n s in o r d e r to f i n d t h e s t a b l e one. In t h e s t r o n g m a g n e t i c c a s e , n? - n~ ~ 1 o r m t - ms ~ 1, t h e e n e r gy d i f f e r e n c e b e t w e e n t h e a n t i p a r a l l e l a n d p a r a l l e l c o n f i g u r a t i o n s of t h e l o c a l i z e d m o m e n t s i s Eft - E , = 2 ( ] V 1 2 1 2 / U [ ( n t / n ~ ) - 2]. T h e p a r allel configuration is thus energetically stable. A t y p i c a l e x a m p l e of r e s u l t s of t h e n u m e r i c a l a n a l y s i s i s p r e s e n t e d i n fig. 1. A n i n t e r e s t i n g conclusion, being at variance with the corresp o n d i n g r e s u l t s of [2] a n d [5], i s t h a t i n g e n e r a l t h e p a r a l l e l c o n f i g u r a t i o n of l o c a l i z e d s p i n s i s favoured energetically. The parallel configurat i o n of l o c a l i z e d m o m e n t s c a n b e c h a n g e d i n t o t h e antiparallel one by increasing z or x.
Reference s 1. P.W. Anderson, Phys. Rev. 124 (1961) 41. 2. S. Alexander and P.W. Anderson, Phys. Rev. 133 (1964) 1594. 3. A. C. Hewson, Phys. Rev. 144 (1966) 420. 4. D.N. Zubarev, Uspekhi Fiz. Nauk, 71 (1960) 71. 5. T. Moriya and M. Inoue, l>rogr. Theor. Phys. 38 (1967) 41.
461
PHYSICS
ion distribution over the magnetic sublattices. T h e d i f f e r e n c e i n e l e c t r o n i c s h e l l s t r u c t u r e of m e t a l i o n s of t h e s e s u b s t a n c e s ( g r o u n d s t a t e s 6S_~ a n d 4F{ ) s e e m s t o b e i m p o r t a n t too. W h i l e t h e 2 d e p e n d e n c e of F e 2 0 3 m a y b e a c c o u n t e d f o r b y t h e m a g n e t o - d i p o l e i n t e r a c t i o n [5], t h e c o r r e s p o n d i n g c a l c u l a t i o n s f o r C r 2 0 3 p r e d i c t a v a l u e [2] w h i c h i s b y a n o r d e r of m a g n i t u d e l e s s t h a n t h e o b s e r v ed one. It c a n b e s u p p o s e d t h a t t h e e f f e c t o b s e r v e d m a y be accounted for by the anisotropic energy contribution connected with the spin-orbit interaction i n t h e c r y s t a l . T h e p r e s e n t t h e o r y i s u n a b l e to produce a quantitative estimate. However, one may expect that in a crystal with a large supere x c h a n g e i n t e r a c t i o n (T N = 3 0 8 ° K ) t h e p r o p e r t i e s of i t s a n i s o t r o p i c p a r t [6] a r e s i g n i f i c a n t in p a r -
EFFECT
OF ELECTRON OF LOCALIZED
LETTERS
15 December 1969
ticular. They are noticeable against the small v a l u e of t o t a l a n i s o t r o p y e n e r g y . We a r e g r a t e f u l t o V. M. F r i d m a n f o r t h e assistance in the experiment. We have benefited f r o m d i s c u s s i o n s w i t h A. S. B o r o v i k - R o m a n o v .
References 1. s. F o n e r and S. L. Hou. J Appl. Phys• 33 (1962) 1289S. 2. J O. Artman. J. C Murphy and S. Foner. Phys. Rev. 138 {1965) 912• 3. J. S. J a c c b s and P. E. Lawrence. Rev. Sei. Instr• 29 (1958) 713. 4. T.G. Worlton. R. M. B r u g g e r and R. B. Bennion. J. Phys• Chem. Sol. 29 (1968) 435. 5. P.A. Voskanian. R. Z. Lewitin and V A Shchurov, Zh. Eksp. i. Teor. Fiz. 54 {1968) 790. 6. T. Moriya, Phys. Rev. 120 {1960) 91.
CORRELATIONS MAGNETIC
ON INTERACTION MOMENTS
J. M O R K O W S K I a n d A. S T A S C H Ferroraagnetics Laborator3.,, Institute of Fhysics of the Polish Academy of Sciences, Pozna~, Poland Received 3 November 1969
R is shown that the effect of e l e c t r o n c o r r e l a t i o n s is to change the c r i t e r i o n for o c c u r r e n c e of localized m a g n e t i c states. The conditions of existence a r e different for p a r a l l e l and a n t i p a r a l l e l configurations of a p a i r of localized magnetic states. In g e n e r a l , the p a r a l l e l configuration is favoured•
W e c o n s i d e r t h e A n d e r s o n [1,2] m o d e l of two impurity transition metal atoms, each having a single, non-degenerate d-state, introduced into a m a t r i x of a n o n - m a g n e t i c m e t a l . T h e s y s t e m of c o n d u c t i o n a n d d - e l e c t r o n s i s d e t e r m i n e d b y t h e t i a m i l t o n i a n ( t h e n o t a t i o n of r e f . 2 i s u s e d throughout).
~g=k~aCknko. +~. E o n i a $a
+ (~ Vikq + ika Cka. C~a r e f e r Ct*a c o r r e s p o n d n -ka - = C k+a C ka ,
+
_t
~. 2Vnicrni,-~r + ,a
(1)
Ck~ + ~ v 1 2 C ~ C 2 ~ + c . c . ) .
t o t h e c o n d u c t i o n e l e c t r o n s , Ci , to the localized d-electrons an~ n -,~ = C ,-+~ , a•" Ek i s t h e e n e r g y of c o n d u c t i o n e l e c t r o n s , E o d e n o t e s t h e e n e r g y of the localized d-state, U is the Coulomb interact i o n e n e r g y ; Vik a n d V12 a r e t h e m a t r i x e l e m e n t s for t h e m i x i n g i n t e r a c t i o n of e l e c t r o n s . W e assume V >> I V/k l, IV121-
46O
T h e e x p e c t a t i o n v a l u e s (n/a~" a r e c a l c u l a t e d by t h e G r e e n f u n c t i o n m e t h o d [4,3]• T h e c h a i n of t h e e q u a t i o n s of m o t i o n f o r t h e G r e e n f u n c t i o n s (( Cicr; C~)) i s r e d u c e d to a c l o s e d s y s t e m b y • . + n e g l e c t i n g t h e G r e e n f u n c t i o n s ((Ck -a Ci~ Ci ~; 4: + + + 7C~.~) a n d (( C~ _~ C~.~ C, _~,; C , ~ ) (ct. [3]), a n d b y th~e~ f o l l o w i n g ~ d e ~ o u ~ l i @ s Y o-
+ =
forj
¢ i.
IV12 ] we f i n d f i n a l l y
(nia) _ 1 - (ni,_~} x n Vrctan ×
La
(2)
EF - E 0 - ( 1 -
-
Volume 30A, n u m b e r 8
PHYSICS
i
. .o . . , . ,
f2 r;g 1o
f3
-Of
N
Fi~ ~b
I
I
I
1
2
3
I
X: A Fig. la. Regions of existence of magnetic and n o n m a g netic solutions of eq. (2). The dotted c u r v e c o r r e s p o n d s to n o n - i n t e r a c t i n g i m p u r i t i e s (z = 0), the two other a r e calculated for z = 0.1. In the region above the full line t h e r e a p p e a r localized magnetic m o m e n t s coupled p a r allel; s i m i l a r l y the b r o k e n line d e t e r m i n e s the domain of existence for a n t i p a r a l l e l configuration of localized magnetic m o m e n t s . Fig. lb. Relative stability of the p a r a l l e l and a n t i p a r allel configurations of localized magnetic m o m e n t s . The two c u r v e s of the e n e r g y difference E . - E , v e r s u s the F e r m i energy or the p a r a m e t e r x w e r e calculated for y = 2.25 and 3.0, respectively. H e r e ( i , j ) = ( 1 , 2 ) o r (2, 1); E F d e n o t e s t h e F e r m i ener~, a n d ~ a n d ,x a r e d e f i n e d b y ~ - iA = = ~ k [ 1 V/k t 2 / ( E F - Ek + i0+)] • (We a s s u m e t h a t a n d ,x a r e s l o w l y v a r y i n g f u n c t i o n s of e n e r g y a n d we t a k e t h e i r v a l u e s f o r E = E F ) . F o l l o w i n g r e f . 2 we s e e k f o r t h e s e l f - c o n s i s t e n t s o l u t i o n s
LETTERS
15 D e c e m b e r 1969
( n l ~ } = ( n 2 a ) = n c r a n d ( n l a > = ( n 2 , _ ~ = m a of eq. (2~, c o r r e s p o n d i n g to p a r a l l e l a n d a n t i p a r a l l e l c o u p l i n g of t h e two l o c a l i z e d m o m e n t s , r e s p e c t i v e l y . T h e m a g n e t i c s o l u t i o n s h a v e to s a t i s f y t h e c o n d i t i o n s n~ - n; ¢ 0 o r m~ - m; ¢ 0. T h e r e s u l t s of t h e t h e o r y a r e d i s c u s s e d in t e r m s of t h e f o l l o w i n g d i m e n s i o n l e s s p a r a m e t e r s : x = ( E o - E F ) / A , y = -~,/A, z = IV12 12,yUA. F r o m r e f . 2 t h e r e g i o n s of t h e s e p a r a m e t e r s c a n b e found, for which there appear localized magnetic states coupled paralle] or antiparallel. There are regions for which both configurations are adm i s s i b l e . In s u c h c a s e s it i s n e c e s s a r y to c a l c u l a t e t h e e n e r g y of t h e s e two c o n f i g u r a t i o n s in o r d e r to f i n d t h e s t a b l e one. In t h e s t r o n g m a g n e t i c c a s e , n? - n~ ~ 1 o r m t - ms ~ 1, t h e e n e r gy d i f f e r e n c e b e t w e e n t h e a n t i p a r a l l e l a n d p a r a l l e l c o n f i g u r a t i o n s of t h e l o c a l i z e d m o m e n t s i s Eft - E , = 2 ( ] V 1 2 1 2 / U [ ( n t / n ~ ) - 2]. T h e p a r allel configuration is thus energetically stable. A t y p i c a l e x a m p l e of r e s u l t s of t h e n u m e r i c a l a n a l y s i s i s p r e s e n t e d i n fig. 1. A n i n t e r e s t i n g conclusion, being at variance with the corresp o n d i n g r e s u l t s of [2] a n d [5], i s t h a t i n g e n e r a l t h e p a r a l l e l c o n f i g u r a t i o n of l o c a l i z e d s p i n s i s favoured energetically. The parallel configurat i o n of l o c a l i z e d m o m e n t s c a n b e c h a n g e d i n t o t h e antiparallel one by increasing z or x.
Reference s 1. P.W. Anderson, Phys. Rev. 124 (1961) 41. 2. S. Alexander and P.W. Anderson, Phys. Rev. 133 (1964) 1594. 3. A. C. Hewson, Phys. Rev. 144 (1966) 420. 4. D.N. Zubarev, Uspekhi Fiz. Nauk, 71 (1960) 71. 5. T. Moriya and M. Inoue, l>rogr. Theor. Phys. 38 (1967) 41.
461