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Effect of superconductivity on the formation of localized magnetic moments
Volume 38A, number 5
PHYSICS LETTERS
EFFECT
28 February 1972
OF SUPERCONDUCTIVITY ON THE FORMATION OF LOCALIZED MAGNETIC MOMENTS
J. R O S S L E R and M. KIWI Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile
Received 12 January 1972 The effect of the superconducting state on the formation of localized moments is investigated. It is shown that the magnetic region is slightly reduced relative to the normal state: a quantitative expression for this reduction is given.
The influence of s u p e r c o n d u c t i v i t y on the f o r m a t i o n of m a g n e t i c m o m e n t s in s u p e r c o n d u c t i n g a l l o y s h a s been studied by v a r i o u s a u t h o r s . Akhytamov and F e d o r o v [1] and T r i p a t h i [2] i n v e s t i g a t e d the s u b j e c t , within the f r a m e w o r k of Z u c k e r m a n n ' s e a r l y w o r k [3] (i.e. without i n t r o d u c i n g the effect of "induced p a i r i n g " [4,5]) and concluded that s u p e r c o n d u c t i v i t y h i n d e r s the m o m e n t f o r m a t i o n . K u s a k a b e [6] included the p a i r i n g induced on the l o c a l i z e d d s t a t e s of the i m p u r i t y [4,5] and concluded that s u p e r conductivity e i t h e r aided o r h i n d e r e d the f o r m a t i o n of l o c a l m o m e n t s a c c o r d i n g to the p o s i t i o n of the d l e v e l r e l a t i v e to the F e r m i s u r f a c e . H o w e v e r , none of the above mentioned a u t h o r s gave a quantitative e s t i m a t e of the s i z e of the effect n o r handled the p r o b l e m s e l f - c o n s i s t e n t l y . In this l e t t e r we r e p o r t q u a n t i t a t i v e r e s u l t s of a s e l f - c o n s i s t e n t t r e a t m e n t of this p r o b l e m (including the effect of "induced p a i r i n g " ) , obtained through the a p p l i c a t i o n of A n d e r s o n ' s c r i t e r i a [7] f o r the f o r m a t i o n of l o c a l i z e d magnetic moments. Using the notation of ref. [2] we obtain f o r the d e l e c t r o n double t i m e G r e e n function [8], in the G o r kov a p p r o x i m a t i o n [9] I 1
G d (¢0) = {w(1 + r / ~ )
+Urn +Ed-~ 3 - CAd + F A / ~ -
¢o2)~1}
(1)
w h e r e A i s the s u p e r c o n d u c t i n g o r d e r p a r a m e t e r , 2m = - (d~£ d j t } , A d = U and E d = E - ½U((d~T d j r ) + ( d ~ d j l ) ) . Using the p r o c e d u r e outlined by Z u b a r e v [8t we can e v a l u a t e n~ =
Ar2urn[r2+ (urn +Ed)2]sgn[r2 +E2 - U2rn 2] n~=larccoti(Urn-FEd)
+
[(F2+E2_U2rn2)2+4F2U2rn213/2
+ 2~-~ddlF2+(Ed-Urn)2 +~L arctanl~ 2r2vm -
F
) +arctan(
(E d - Urn )(r 2 +E~ - v2rn 2)
F
]I
).
The analogous equation f o r n T i s s i m p l y obtained by l e t t i n g rn -. - m in r e l a t i o n (2); t h e s e two equations, f o r n~ and nT, f o r m a s e l f - c o n s i s t e n t s e t which we s o l v e f o r the l i m i t i n g c a s e of m --. 0 (i.e. we find the m a g n e t i c - non m a g n e t i c t r a n s i t i o n line). Using the p a r a m e t e r s ~7 = ~ F / U , x = E / U and c a l l i n g ~7o(X) the n o r m a l s t a t e solution obtained by A n d e r s o n [7] we find 371
Volume 38A, number 5
PHYSICS
LETTERS
28 February 1972
v(x) - Vo(X) = -,~Vo(X) -6x~I ln~] ~2o(X) - ½,x 2 Ix- ~ ][Vo(X)/(1 - ~o(X))] 1/2 ,
(3)
w h e r e the d e f i n i t i o n s 6 = A / r and )L = A d / F h a v e been u s e d . S i n c e i n g e n e r a l 6 ~ 10 -3 and ~ ~ 10 -2 the m i n i m u m v a l u e of ~(x) - ~?o(X) o c c u r s a t x = ½ and in m a g n i t u d e is -()t2 + u 6 + 6 ~ 6 [ l n 6 I) ~ 10 -3. T h e m i n u s s i g n i n d i c a t e s that the m a g n e t i c r e g i o n i s r e d u c e d in r e l a t i o n to the n o r m a l s t a t e r e s u l t s . On the o t h e r hand, w h e n one u s e s the m e t h o d s of r e f . [7], the c o n c l u s i o n is r e a c h e d that an e x i s t i n g m a g n e t i c m o m e n t i s e n h a n c e d by s u p e r c o n d u c t i v i t y if U 2 m ~ > E 2d + r 2 and is r e d u c e d o t h e r w i s e . It i s i m p o r t a n t to e m p h a s i z e that the s i z e of the r e d u c t i o n of the m a g n e t i c r e g i o n is v e r y s m a l l , m a k i n g the e x p e r i m e n t a l o b s e r v a t i o n of the e f f e c t e x t r e m e l y u n l i k e l y . M o r e o v e r , f r o m a p u r e l y t h e o r e t i c a l p o i n t of v i e w , the p i c t u r e of a s h a r p t r a n s i t i o n h a s b e e n s e v e r e l y c r i t i c i z e d ~10] and a m o r e r e a l i s t i c c o n c e p t i o n , with a s m o o t h t r a n s i t i o n b e t w e e n the m a g n e t i c and n o n - m a g n e t i c r e g i m e s h a s e m e r g e d [11]. T h i s f u r t h e r s t r e n g t h e n s o u r point, s i n c e the c o r r e c t i o n due to the s u p e r c o n d u c t i n g s t a t e i s m u c h n a r r o w e r than the i n t e r m e d i a t e r e g i o n b e t w e e n the m a g n e t i c and n o n m a g n e t i c r e g i m e s .
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
372
o. s. Akhtyamov and E. I. Fedorov, J E T P Letters 4 (1966) 280. R. S. Tripathi, Phys. Letters 25A (1967) 381. M. J. Zuckermann, Phys. Rev. 140 (1965) 899. C. F. Hatto and A. Blandin, Phys. Rev. 156 (1967) 513. M. Kiwi and M. J. Zuckermann, Phys. Rev. 164 (1967) 548. T. Kusakabe, Progr. Theor. Phys. 45 (1971) 651. P. W. Anderson, Phys. Rev. 124 (1961) 41. D. N. Zubarev, Sov. Phys. Uspekhi 3 (1960) 320. L. P. Gorkov, Zh. Eksp. i T e o r . Fiz. 7 (1958) 505. H. Suhl, Phys. Rev. Letters 19 (1967) 442. H. Keiter, Conf. on the Electric and magnetic properties of dilute alloys (Tihany, Hungary, 1971).
PHYSICS LETTERS
EFFECT
28 February 1972
OF SUPERCONDUCTIVITY ON THE FORMATION OF LOCALIZED MAGNETIC MOMENTS
J. R O S S L E R and M. KIWI Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile
Received 12 January 1972 The effect of the superconducting state on the formation of localized moments is investigated. It is shown that the magnetic region is slightly reduced relative to the normal state: a quantitative expression for this reduction is given.
The influence of s u p e r c o n d u c t i v i t y on the f o r m a t i o n of m a g n e t i c m o m e n t s in s u p e r c o n d u c t i n g a l l o y s h a s been studied by v a r i o u s a u t h o r s . Akhytamov and F e d o r o v [1] and T r i p a t h i [2] i n v e s t i g a t e d the s u b j e c t , within the f r a m e w o r k of Z u c k e r m a n n ' s e a r l y w o r k [3] (i.e. without i n t r o d u c i n g the effect of "induced p a i r i n g " [4,5]) and concluded that s u p e r c o n d u c t i v i t y h i n d e r s the m o m e n t f o r m a t i o n . K u s a k a b e [6] included the p a i r i n g induced on the l o c a l i z e d d s t a t e s of the i m p u r i t y [4,5] and concluded that s u p e r conductivity e i t h e r aided o r h i n d e r e d the f o r m a t i o n of l o c a l m o m e n t s a c c o r d i n g to the p o s i t i o n of the d l e v e l r e l a t i v e to the F e r m i s u r f a c e . H o w e v e r , none of the above mentioned a u t h o r s gave a quantitative e s t i m a t e of the s i z e of the effect n o r handled the p r o b l e m s e l f - c o n s i s t e n t l y . In this l e t t e r we r e p o r t q u a n t i t a t i v e r e s u l t s of a s e l f - c o n s i s t e n t t r e a t m e n t of this p r o b l e m (including the effect of "induced p a i r i n g " ) , obtained through the a p p l i c a t i o n of A n d e r s o n ' s c r i t e r i a [7] f o r the f o r m a t i o n of l o c a l i z e d magnetic moments. Using the notation of ref. [2] we obtain f o r the d e l e c t r o n double t i m e G r e e n function [8], in the G o r kov a p p r o x i m a t i o n [9] I 1
G d (¢0) = {w(1 + r / ~ )
+Urn +Ed-~ 3 - CAd + F A / ~ -
¢o2)~1}
(1)
w h e r e A i s the s u p e r c o n d u c t i n g o r d e r p a r a m e t e r , 2m =
Ar2urn[r2+ (urn +Ed)2]sgn[r2 +E2 - U2rn 2] n~=larccoti(Urn-FEd)
+
[(F2+E2_U2rn2)2+4F2U2rn213/2
+ 2~-~ddlF2+(Ed-Urn)2 +~L arctanl~ 2r2vm -
F
) +arctan(
(E d - Urn )(r 2 +E~ - v2rn 2)
F
]I
).
The analogous equation f o r n T i s s i m p l y obtained by l e t t i n g rn -. - m in r e l a t i o n (2); t h e s e two equations, f o r n~ and nT, f o r m a s e l f - c o n s i s t e n t s e t which we s o l v e f o r the l i m i t i n g c a s e of m --. 0 (i.e. we find the m a g n e t i c - non m a g n e t i c t r a n s i t i o n line). Using the p a r a m e t e r s ~7 = ~ F / U , x = E / U and c a l l i n g ~7o(X) the n o r m a l s t a t e solution obtained by A n d e r s o n [7] we find 371
Volume 38A, number 5
PHYSICS
LETTERS
28 February 1972
v(x) - Vo(X) = -,~Vo(X) -6x~I ln~] ~2o(X) - ½,x 2 Ix- ~ ][Vo(X)/(1 - ~o(X))] 1/2 ,
(3)
w h e r e the d e f i n i t i o n s 6 = A / r and )L = A d / F h a v e been u s e d . S i n c e i n g e n e r a l 6 ~ 10 -3 and ~ ~ 10 -2 the m i n i m u m v a l u e of ~(x) - ~?o(X) o c c u r s a t x = ½ and in m a g n i t u d e is -()t2 + u 6 + 6 ~ 6 [ l n 6 I) ~ 10 -3. T h e m i n u s s i g n i n d i c a t e s that the m a g n e t i c r e g i o n i s r e d u c e d in r e l a t i o n to the n o r m a l s t a t e r e s u l t s . On the o t h e r hand, w h e n one u s e s the m e t h o d s of r e f . [7], the c o n c l u s i o n is r e a c h e d that an e x i s t i n g m a g n e t i c m o m e n t i s e n h a n c e d by s u p e r c o n d u c t i v i t y if U 2 m ~ > E 2d + r 2 and is r e d u c e d o t h e r w i s e . It i s i m p o r t a n t to e m p h a s i z e that the s i z e of the r e d u c t i o n of the m a g n e t i c r e g i o n is v e r y s m a l l , m a k i n g the e x p e r i m e n t a l o b s e r v a t i o n of the e f f e c t e x t r e m e l y u n l i k e l y . M o r e o v e r , f r o m a p u r e l y t h e o r e t i c a l p o i n t of v i e w , the p i c t u r e of a s h a r p t r a n s i t i o n h a s b e e n s e v e r e l y c r i t i c i z e d ~10] and a m o r e r e a l i s t i c c o n c e p t i o n , with a s m o o t h t r a n s i t i o n b e t w e e n the m a g n e t i c and n o n - m a g n e t i c r e g i m e s h a s e m e r g e d [11]. T h i s f u r t h e r s t r e n g t h e n s o u r point, s i n c e the c o r r e c t i o n due to the s u p e r c o n d u c t i n g s t a t e i s m u c h n a r r o w e r than the i n t e r m e d i a t e r e g i o n b e t w e e n the m a g n e t i c and n o n m a g n e t i c r e g i m e s .
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
372
o. s. Akhtyamov and E. I. Fedorov, J E T P Letters 4 (1966) 280. R. S. Tripathi, Phys. Letters 25A (1967) 381. M. J. Zuckermann, Phys. Rev. 140 (1965) 899. C. F. Hatto and A. Blandin, Phys. Rev. 156 (1967) 513. M. Kiwi and M. J. Zuckermann, Phys. Rev. 164 (1967) 548. T. Kusakabe, Progr. Theor. Phys. 45 (1971) 651. P. W. Anderson, Phys. Rev. 124 (1961) 41. D. N. Zubarev, Sov. Phys. Uspekhi 3 (1960) 320. L. P. Gorkov, Zh. Eksp. i T e o r . Fiz. 7 (1958) 505. H. Suhl, Phys. Rev. Letters 19 (1967) 442. H. Keiter, Conf. on the Electric and magnetic properties of dilute alloys (Tihany, Hungary, 1971).