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Low temperature specific heat of dilute magnetic alloys
Volume 24.A, number 2
LOW
PHYSICS LETTERS
TEMPERATURE
SPECIFIC
HEAT
16 January 1967
OF DILUTE
MAGNETIC
ALLOYS
D. J. KIM and B. B. SCHWARTZ
National Magnet Laboratory ~, Massachusetts Institue of Technology Cambridge, Massachusetts Received 12 December 1966
The effective ferromagnetic exchange interaction between conduction electrons produced by the presence of magnetic impurities is shown to increase the electronic specific heat by the spin fluctuation mechanism recently presented by Berk and Schrieffer, and Doniach and Engelsberg. The a n o m a l o u s b e h a v i o r of the low t e m p e r a t u r e specific heat of dilute m a g n e t i c alloys has been a c o n t r o v e r s i a l p r o b l e m . If we i n t r o d u c e , for i n s t a n c e , 1 atomic p e r cent of Mn in Cu, then the l i n e a r in T t e r m in the specific heat i n c r e a s e s by 5-6 t i m e s over that of p u r e Cu and this i n c r e a s e is n e a r l y independent of Mn c o n c e n t r a t i o n in the range of ½-10 a t o m i c p e r cent [1]. S i m i l a r b e h a v i o r has been o b s e r v e d g e n e r a l l y when the i n t r o duced i m p u r i t y has a localized m o m e n t . S e v e r a l different explanations have been p r e s e n t e d [2-6] but it s e e m s that we have not yet a r r i v e d at a s a t i s f a c t o r y u n d e r s t a n d i n g e s p e c i a l l y as to the magnitude of the i n c r e a s e . Recently in o r d e r to explain the f e r r o m a g n e t i s m of dilute m a g n e t i c alloys it was shown that the p r e s e n c e of m a g n e t i c i m p u r i t i e s gives r i s e to an effective exchange i n t e r a c t i o n between host m e t a l conduction e l e c t r o n s [7]. By c o m b i n i n g this fact with the s p i n f l u c t u a tion i n t e r a c t i o n r e c e n t l y c o n s i d e r e d by Berk and S c h r i e f f e r [8], and Doniach and E n g e l s b e r g [9] we p r e s e n t h e r e a m e c h a n i s m which can account for the l a r g e e n h a n c e m e n t of the specific heat. According to these a u t h o r s [8, 9] the specific heat of m e t a l l i c e l e c t r o n s is enhanced due to the i n t e r a c t i o n of the individual e l e c t r o n with the spin fluctuation of the e l e c t r o n m e d i u m by the following r e l a t i o n $~t --m - 1 --aN(0) Vc In
a ~
E' 1 1 - N ( 0 ) Vc
2 for N(0) Vc ~ 1
(1)
(2)
1 for N(0) Vc << 1 $ Supported by the U.S.Air Force Office of Scientific Research. $$ Here we follow ref° 8 and neglect the effect of phonons which is not supposed to be affected drastically by the presence of impurities.
where m is the e l e c t r o n m a s s which does not i n clude the effect of spin fluctuation and m* is the effective m a s s which d e t e r m i n e s the e l e c t r o n i c specific heat. N(0) is the density of s t a t e s of e l e c t r o n s of one s p i n at the F e r m i s u r f a c e and Vc is the effective exchange i n t e r a c t i o n by which the s o - c a l l e d exchange e n h a n c e m e n t factor is given as [1 -N(0) Vc]-I. One expects the deviation of m * / m from 1, due to the i n t e r a c t i o n with the spin fluctuation, to become a p p r e c i a b l e when N(0) Vc approaches 1. In p u r e Cu, N(0) Vc is much l e s s than 1 and thus rn*/rn ~ 1. When a s m a l l quantity of Mn is put into Cu, however, an effective f e r r o m a g n e t i c exchange i n t e r a c t i o n is produced among host m e t a l conduction e l e c t r o n s [7]. Or, m o r e d i r e c t l y , it can be shown that the e l e c t r o n spin fluctuation, n a m e l y the dyn a m i c a l spin s u s c e p t i b i l i t y of e l e c t r o n s , is e n hanced by the p r e s e n c e of the localized i m p u r i t y s p i n s [10]. The magnitude of the effective i n t e r action in the static l i m i t , which is to be i n s e r t e d on the r i g h t - h a n d side of eq. (1), is given as [7]
Sz
Veff = 2j2 ~ / g p ~
) .
(3)
Here c is the i m p u r i t y c o n c e n t r a t i o n and equal to
N o / N , the n u m b e r of i m p u r i t i e s p e r unit volume divided by the n u m b e r of a t o m s p e r unit volume; J is the s - d exchange i n t e r a c t i o n between the i m p u r i t y spin (S) and the conduction electron; g ~ B I t S is the Z e e m a n e n e r g y of the localized spin u n d e r the local field due to s u r r o u n d i n g other i m p u r i t y spins; and ( . . . ) m e a n s the t h e r m a l a v e r a g e . At high t e m p e r a t u r e s the b r a c k e t e d quantity in eq.(3) may be r e p l a c e d by the s u s c e p t i b i l i t y of localized spin, which is the C u r i e law; however, at low t e m p e r a t u r e , such as k B T ~ ]gl.tBH I , this b r a c k e t e d quantity b e c o m e s l e s s t e m p e r a t u r e dependent and i n v e r s e l y p r o p o r t i o n a l to c s i n c e 77
Volume 24A, n u m b e r 2
PHYSICS
LETTERS
gttBH ~ c J 2 N ( 0 ) SIN.
T h u s Vef f in e q . ( 3 ) b e comes temperature and concentration independent a t low t e m p e r a t u r e s : Vef f ~ ~N(0) -1
(4)
w h e r e ~ i s a c o n s t a n t of t h e o r d e r 1 [ 5 , 6 ] . In o r d e r to a c c o u n t f o r t h e s p e c i f i c h e a t d a t a of C u - M n , f o r i n s t a n c e , s o l e l y f r o m t h e p r e s e n t m e c h a n i s m ~, it i s r e q u i r e d to a s s u m e )t ~ 0.9 to m a k e m * / m -1 ~ 5 in eq. (1). T h i s v a l u e f o r ~t s e e m s to be w i t h i n a r e a s o n a b l e r a n g e [6]. The effective ferromagnetic interaction which can produce a large excess specific heat, should, by the mechanism considered by Berk and Schrieff e r [8], l e a d to t h e d e s t r u c t i o n of s u p e r c o n d u c t i v i t y . A c t u a l l y , we m a y h a v e s e e n s u c h a c a s e , f o r i n s t a n c e , in T i - M n a l l o y s [11]. We w o u l d l i k e to t h a n k D r . A. J . F r e e m a n his helpful discussions.
for
16 January 1967
References 1. F o r an extensive review on the experimental situation see G. J. Van den Berg, 9th Intern. Conf. on Low t e m p e r a t u r e physics (Plenum P r e s s , New York 1965) 955. 2. W. O v e r h a u s e r , J. Phys. Chem. Solids 13 (1960) 71. 3. W . M a r s h a l l , P h y s . R e v . l l 8 (1960) 15i9. 4. M . W . K l e i n and R . B r o u t , Phys.Rev. 132 (1963) 2412. 5. J.Kondo, P r o g r . T h e o r e t . Phys. (Kyoto) 33 (1965) 575. 6. M.W.Klein, P h y s . R e v . L e t t e r s 16 (1966) 127. 7. D . J . K i m , P h y s . R e v . 149 (1966) 434. Also see M. Bailyn, Adv. in Phys. 15 (1960) 179 and r e f s . 5 and 11. 8. N. F. B c r k and J . R . Schrieffer, P h y s . R e v . L e t t e r s 17 (1966) 433; 10th Intern. Conf. on Low t e m p e r a t u r e physics (Moscow, 1966) to be published; 12th Annual Conf. on Magnetism and magnetic m a t e r i a l s (Washington, 1966); to be published. 9. S. Doniach and S. Engelsberg, P h y s . R e v . L e t t e r s 17 (1966) 750. I0. S. Doniach and E. P. Wohlfarth, Proc.Roy. Soc., to be published. II. R.R.Hakeand J.A,Cape, Phys.Rev.135 (1965)Al151.
$ T h e r e may be other additive contributions, due to such m e c h a n i s m s as d i s c u s s e d in r e f s . 2-5. **~**
FLUX TRAPPING
IN R O T A T I N G S U P E R C O N D U C T O R S *
P. T. SIKORA and K. J. CARROLL
University of California, Los Alarnos Scientific Laboratory, Los Alaraos, New Mexico Received 12 D e c e m b e r 1966
A g e o m e t r i c a l model is proposed to explain the dependence of flux trapping on angular velocity when superconducting Pb and Nb s p h e r e s in an external magnetic field a r e cooled through t h e i r t r a n s i t i o n t e m p e r a t u r e s while in rotation.
S e v e r a l e x p e r i m e n t s on t h e r o t a t i o n of s i m p l y or multiply-connected metal samples undergoing the normal-superconducting transition in a constant magnetic field have been explained with moderate success by simple eddy-current theory ~ 1 , 2 ] . S t u d i e s c a r r i e d o u t b y o n e of u s [3] d i d n o t s e e m to l e n d t h e m s e l v e s to s u c h a n i n t e r p r e t a t i o n . The eddy-current approach was therefore aband o n e d i n f a v o r of a m o r e i n t u i t i v e p h e n o m e n o l o g i cal model. We now d i s c u s s t h e m o d e L a d o p t e d to e x p l a i n t h e d e p e n d e n c e of t h e t r a p p e d f l u x o n t h e s p e e d of 78
rotation. This model is based on the assumptions a) t h a t t h e f l u x t r a p p i n g i s i n f l u e n c e d by t h e p r e s e n c e of l a t t i c e i m p e r f a c t i o n s i n t h e s u p e r c o n d u c t i n g s p h e r e s , b) t h a t t h e d i s t r i b u t i o n of t h e s e imperfections is uniform and isotropic throughout t h e s p h e r e , a n d c) t h a t f l u x t r a p p i n g a t a g i v e n s i t e m a y o c c u r o n l y if t h e t e n t a t i v e p a t h f o r t h e trapped flux line remains aligned with the magn e t i c f i e l d f o r s o m e m i n i m u m t i m e ~. * Work p e r f o r m e d u n d e r the auspices of the U. S, Atomic Energy C o m m i s s i o n .
LOW
PHYSICS LETTERS
TEMPERATURE
SPECIFIC
HEAT
16 January 1967
OF DILUTE
MAGNETIC
ALLOYS
D. J. KIM and B. B. SCHWARTZ
National Magnet Laboratory ~, Massachusetts Institue of Technology Cambridge, Massachusetts Received 12 December 1966
The effective ferromagnetic exchange interaction between conduction electrons produced by the presence of magnetic impurities is shown to increase the electronic specific heat by the spin fluctuation mechanism recently presented by Berk and Schrieffer, and Doniach and Engelsberg. The a n o m a l o u s b e h a v i o r of the low t e m p e r a t u r e specific heat of dilute m a g n e t i c alloys has been a c o n t r o v e r s i a l p r o b l e m . If we i n t r o d u c e , for i n s t a n c e , 1 atomic p e r cent of Mn in Cu, then the l i n e a r in T t e r m in the specific heat i n c r e a s e s by 5-6 t i m e s over that of p u r e Cu and this i n c r e a s e is n e a r l y independent of Mn c o n c e n t r a t i o n in the range of ½-10 a t o m i c p e r cent [1]. S i m i l a r b e h a v i o r has been o b s e r v e d g e n e r a l l y when the i n t r o duced i m p u r i t y has a localized m o m e n t . S e v e r a l different explanations have been p r e s e n t e d [2-6] but it s e e m s that we have not yet a r r i v e d at a s a t i s f a c t o r y u n d e r s t a n d i n g e s p e c i a l l y as to the magnitude of the i n c r e a s e . Recently in o r d e r to explain the f e r r o m a g n e t i s m of dilute m a g n e t i c alloys it was shown that the p r e s e n c e of m a g n e t i c i m p u r i t i e s gives r i s e to an effective exchange i n t e r a c t i o n between host m e t a l conduction e l e c t r o n s [7]. By c o m b i n i n g this fact with the s p i n f l u c t u a tion i n t e r a c t i o n r e c e n t l y c o n s i d e r e d by Berk and S c h r i e f f e r [8], and Doniach and E n g e l s b e r g [9] we p r e s e n t h e r e a m e c h a n i s m which can account for the l a r g e e n h a n c e m e n t of the specific heat. According to these a u t h o r s [8, 9] the specific heat of m e t a l l i c e l e c t r o n s is enhanced due to the i n t e r a c t i o n of the individual e l e c t r o n with the spin fluctuation of the e l e c t r o n m e d i u m by the following r e l a t i o n $~t --m - 1 --aN(0) Vc In
a ~
E' 1 1 - N ( 0 ) Vc
2 for N(0) Vc ~ 1
(1)
(2)
1 for N(0) Vc << 1 $ Supported by the U.S.Air Force Office of Scientific Research. $$ Here we follow ref° 8 and neglect the effect of phonons which is not supposed to be affected drastically by the presence of impurities.
where m is the e l e c t r o n m a s s which does not i n clude the effect of spin fluctuation and m* is the effective m a s s which d e t e r m i n e s the e l e c t r o n i c specific heat. N(0) is the density of s t a t e s of e l e c t r o n s of one s p i n at the F e r m i s u r f a c e and Vc is the effective exchange i n t e r a c t i o n by which the s o - c a l l e d exchange e n h a n c e m e n t factor is given as [1 -N(0) Vc]-I. One expects the deviation of m * / m from 1, due to the i n t e r a c t i o n with the spin fluctuation, to become a p p r e c i a b l e when N(0) Vc approaches 1. In p u r e Cu, N(0) Vc is much l e s s than 1 and thus rn*/rn ~ 1. When a s m a l l quantity of Mn is put into Cu, however, an effective f e r r o m a g n e t i c exchange i n t e r a c t i o n is produced among host m e t a l conduction e l e c t r o n s [7]. Or, m o r e d i r e c t l y , it can be shown that the e l e c t r o n spin fluctuation, n a m e l y the dyn a m i c a l spin s u s c e p t i b i l i t y of e l e c t r o n s , is e n hanced by the p r e s e n c e of the localized i m p u r i t y s p i n s [10]. The magnitude of the effective i n t e r action in the static l i m i t , which is to be i n s e r t e d on the r i g h t - h a n d side of eq. (1), is given as [7]
Sz
Veff = 2j2 ~ / g p ~
) .
(3)
Here c is the i m p u r i t y c o n c e n t r a t i o n and equal to
N o / N , the n u m b e r of i m p u r i t i e s p e r unit volume divided by the n u m b e r of a t o m s p e r unit volume; J is the s - d exchange i n t e r a c t i o n between the i m p u r i t y spin (S) and the conduction electron; g ~ B I t S is the Z e e m a n e n e r g y of the localized spin u n d e r the local field due to s u r r o u n d i n g other i m p u r i t y spins; and ( . . . ) m e a n s the t h e r m a l a v e r a g e . At high t e m p e r a t u r e s the b r a c k e t e d quantity in eq.(3) may be r e p l a c e d by the s u s c e p t i b i l i t y of localized spin, which is the C u r i e law; however, at low t e m p e r a t u r e , such as k B T ~ ]gl.tBH I , this b r a c k e t e d quantity b e c o m e s l e s s t e m p e r a t u r e dependent and i n v e r s e l y p r o p o r t i o n a l to c s i n c e 77
Volume 24A, n u m b e r 2
PHYSICS
LETTERS
gttBH ~ c J 2 N ( 0 ) SIN.
T h u s Vef f in e q . ( 3 ) b e comes temperature and concentration independent a t low t e m p e r a t u r e s : Vef f ~ ~N(0) -1
(4)
w h e r e ~ i s a c o n s t a n t of t h e o r d e r 1 [ 5 , 6 ] . In o r d e r to a c c o u n t f o r t h e s p e c i f i c h e a t d a t a of C u - M n , f o r i n s t a n c e , s o l e l y f r o m t h e p r e s e n t m e c h a n i s m ~, it i s r e q u i r e d to a s s u m e )t ~ 0.9 to m a k e m * / m -1 ~ 5 in eq. (1). T h i s v a l u e f o r ~t s e e m s to be w i t h i n a r e a s o n a b l e r a n g e [6]. The effective ferromagnetic interaction which can produce a large excess specific heat, should, by the mechanism considered by Berk and Schrieff e r [8], l e a d to t h e d e s t r u c t i o n of s u p e r c o n d u c t i v i t y . A c t u a l l y , we m a y h a v e s e e n s u c h a c a s e , f o r i n s t a n c e , in T i - M n a l l o y s [11]. We w o u l d l i k e to t h a n k D r . A. J . F r e e m a n his helpful discussions.
for
16 January 1967
References 1. F o r an extensive review on the experimental situation see G. J. Van den Berg, 9th Intern. Conf. on Low t e m p e r a t u r e physics (Plenum P r e s s , New York 1965) 955. 2. W. O v e r h a u s e r , J. Phys. Chem. Solids 13 (1960) 71. 3. W . M a r s h a l l , P h y s . R e v . l l 8 (1960) 15i9. 4. M . W . K l e i n and R . B r o u t , Phys.Rev. 132 (1963) 2412. 5. J.Kondo, P r o g r . T h e o r e t . Phys. (Kyoto) 33 (1965) 575. 6. M.W.Klein, P h y s . R e v . L e t t e r s 16 (1966) 127. 7. D . J . K i m , P h y s . R e v . 149 (1966) 434. Also see M. Bailyn, Adv. in Phys. 15 (1960) 179 and r e f s . 5 and 11. 8. N. F. B c r k and J . R . Schrieffer, P h y s . R e v . L e t t e r s 17 (1966) 433; 10th Intern. Conf. on Low t e m p e r a t u r e physics (Moscow, 1966) to be published; 12th Annual Conf. on Magnetism and magnetic m a t e r i a l s (Washington, 1966); to be published. 9. S. Doniach and S. Engelsberg, P h y s . R e v . L e t t e r s 17 (1966) 750. I0. S. Doniach and E. P. Wohlfarth, Proc.Roy. Soc., to be published. II. R.R.Hakeand J.A,Cape, Phys.Rev.135 (1965)Al151.
$ T h e r e may be other additive contributions, due to such m e c h a n i s m s as d i s c u s s e d in r e f s . 2-5. **~**
FLUX TRAPPING
IN R O T A T I N G S U P E R C O N D U C T O R S *
P. T. SIKORA and K. J. CARROLL
University of California, Los Alarnos Scientific Laboratory, Los Alaraos, New Mexico Received 12 D e c e m b e r 1966
A g e o m e t r i c a l model is proposed to explain the dependence of flux trapping on angular velocity when superconducting Pb and Nb s p h e r e s in an external magnetic field a r e cooled through t h e i r t r a n s i t i o n t e m p e r a t u r e s while in rotation.
S e v e r a l e x p e r i m e n t s on t h e r o t a t i o n of s i m p l y or multiply-connected metal samples undergoing the normal-superconducting transition in a constant magnetic field have been explained with moderate success by simple eddy-current theory ~ 1 , 2 ] . S t u d i e s c a r r i e d o u t b y o n e of u s [3] d i d n o t s e e m to l e n d t h e m s e l v e s to s u c h a n i n t e r p r e t a t i o n . The eddy-current approach was therefore aband o n e d i n f a v o r of a m o r e i n t u i t i v e p h e n o m e n o l o g i cal model. We now d i s c u s s t h e m o d e L a d o p t e d to e x p l a i n t h e d e p e n d e n c e of t h e t r a p p e d f l u x o n t h e s p e e d of 78
rotation. This model is based on the assumptions a) t h a t t h e f l u x t r a p p i n g i s i n f l u e n c e d by t h e p r e s e n c e of l a t t i c e i m p e r f a c t i o n s i n t h e s u p e r c o n d u c t i n g s p h e r e s , b) t h a t t h e d i s t r i b u t i o n of t h e s e imperfections is uniform and isotropic throughout t h e s p h e r e , a n d c) t h a t f l u x t r a p p i n g a t a g i v e n s i t e m a y o c c u r o n l y if t h e t e n t a t i v e p a t h f o r t h e trapped flux line remains aligned with the magn e t i c f i e l d f o r s o m e m i n i m u m t i m e ~. * Work p e r f o r m e d u n d e r the auspices of the U. S, Atomic Energy C o m m i s s i o n .