Superconductivity of dilute magnetic alloys at low temperature

Superconductivity of dilute magnetic alloys at low temperature

Volume 18, number 2 PHYSICS LETTERS the s i l v e r m o l e c u l u r volume t u r n s out to be equal to 1.7 X 10 -23 c m 3. M o r e c o m p l i c ...

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Volume 18, number 2

PHYSICS LETTERS

the s i l v e r m o l e c u l u r volume t u r n s out to be equal to 1.7 X 10 -23 c m 3. M o r e c o m p l i c a t e d i s the deduction of the t r u e value belonging to the r a t i o p / p ' , which r e q u i r e s the c a l c u l a t i o n of the s c r e w d i s l o c a t i o n d e n s i t y , s i n c e t h e s e l a t t i c e i m p e r f e c t i o n s a r e a b l e to g r e a t l y f a c i l i t a t e the p r o c e s s of c r y s t a l g r o w t h d u r i n g the a c c u m u l a t i o n of the a t o m s a r o u n d the p r i m o g e n i a l c o n d e n s a t i o n c e n t r e s which the t h e r m a l a g i t a t i o n c r e a t e s within the f i l m [7]. The n u m b e r of s u c h Une d e f e c t s p a s s i n g on an average through a generic 1 cm 2 cross section i s about 5.5 × 1010. T h i s value h a s b e e n d e t e r m i n e d c o n s i d e r i n g that during the f i r s t p h a s e of t h e i r constitution the s i l v e r c r y s t a l l i n e n u c l e i r e s u l t v e r y d e f o r m e d , b e c a u s e they a r e s u b j e c t e d to i n t e r n a l t e n s i o n s p r o d u c e d by s t a b i l i z i n g o s c i U a t i o n s and to c o m p r e s s i o n s e x e r c i s e d by the n e a r e s t e m b r y o s , s o that the e n e r g y c o n n e c t e d to d i s t o r s i o n s and s l i p p i n g s s t o r e d in the l a t t i c e i s equal to about 2 × 108 e r g / c m 3 . Therefore, so that a primogenial germ crea t e d by s t a t i s t i c a l f l u c t u a t i o n s can g r o w through a s p i r a l o r loop d e p o s i t i o n m e c h a n i s m , a v a p o u r s u p e r s a t u r a t i o n c o r r e s p o n d i n g to the value p/p' = 1.01 .uill be s u f f i c i e n t . I n t r o d u c i n g the j u s t r e l i o r t e d v a l u e s into eq. (3), we can o b t a i n the r e s u l t s shown in t a b l e 1. Since the t h r e s h o l d m e a n r a d i u s Y depends by T through a r e l a t i o n of i n v e r s e p r o p o r t i o n a l i t y , we conclude by o b s e r v i n g that, if a l l the r e m a i n ing conditions a r e k e p t unchanged, the f i l m c r y s tallization becomes gradually always easier as the c o n d e n s a t i o n t e m p e r a t u r e i n c r e a s e s , p r o v i d e d

15 August 1965 Table 1

Condensation

Mean critical radius

temperature (°C)

of the embryon (~)

200 300 400 500 600 700 800 9OO 960.5 (meRing poin~

3836 3166 2696

2347 2078 1865 1691 1547 1471

that this last is not too near the meRing point, where the great thermal agitation hinders the disorder-order transition necessary for the formation of steady crystals. The results of microscopic and diffractoscopic analysis executed in our laboratory on vacuum de. posited silver coatings employed as specular surfaces and electrical leads agree satisfactorily with these theoretical previsions.

References 1. M.Volmer and A. Weber, Z. Phys. Chem. A l l 9 (1926) 277. 2. H.Eyring, J.Chem.Phys.3 (1934) 107. 3. R. Becket, Disc. Faraday Sec. 5 (1949) 55. 4. I.Stranski, S.B.Akad.Wiss.Kl.II b; 145 (1936) 840. 5. M.Von Laue, Z.Kxist.105 (1943) 124. 6. S . W . S m i t h , J. Inst. Met. 12 (1914) 168.

7. F . C . Frank, Adv. Phys. 1 (1952) 91.

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SUPERCONDUCTIVITY OF DILUTE MAGNETIC AT LOW TEMPERATURE

ALLOYS

S. N A K A J I M A *

Cavendish Laboratwry , University of Cambridge, Cambridge, England Received 23 July 1965

We have e x t e n d e d the A b r i k o s o v - G o r ' k o v t h e o r y [I] on s u p e r c o n d u c t i n g m a g n e t i c a l l o y s to i n clude the c a s e of low s p i n t e m p e r a t u r e . T h i s * Permanent ~ : M ~ for Solid 8tare P h y s i c s , U n i v e r s i t y of Tokyo, Azalm, Tokyo.

m e a n s that i m p u r i t y s p i n s a r e no l o n g e r f r e e to r o t a t e b e c a u s e of the i n t e r a c t i o n between t h e m (or m o r e s i m p l y b e c a u s e of a c r y s t a l a a l s o t r o p y ) . The m e t h o d u s e d i s a s i m p l e g e n e r a l i z a t i o n of E14--hberg' s t h e o r y [2] of the e l e c t r o n - p h o n o n coupling, to which we a d d the e x c h a n g e couplin .~ 107

Volume 18, number 2

PHYSICS L E T T E R S

of conduction e l e c t r o n s with l o c a l i z e d i m p u r i t y spins (the s o - c a l l e d sd model). Equations of motion for o n e - e l e c t r o n Green' s functions a r e decoupled by applying Migdal's approximation [3] to electron-phonon as well as to e l e c t r o n - s p i n t h r e e - v e r t e x functions. Thus, in addition to the phonon propagator, there a p p e a r s the "spin propagator", which actually d e s c r i b e s the dynamical c o r r e l a t i o n of i m p u r i t y spins. The s i m p l e s t approximation for the spin propagator is the "Einstein model". We thus a s s u m e that the impurity spin has a c e r t a i n Zeeman ene r g y co2 due to the m o l e c u l a r field a r i s i n g f r o m the interaction with other spins. We also a s s u m e that the orientation of the molecular field is r a n dom, so that there is no long range spin o r d e r i n g on the average. Then the spin p r o p a g a t o r takes the form

~u(~)=~

2

c~- ~ 2 + i 0 +

cu +c~2- i0+

.

(1)

Here ~ is the d i m e n s i o n l e s s coupling p a r a m e t e r given by (SniNoSJ2/c~2) with the number density of impurity atoms hi, the magnitude of the i m purity spin S, the sd exchange coupling constant J, and the density of o n e - e l e c t r o n s t a t e s at the F e r m i surface, N o. In a n o r m a l alloy, the spin p r o p a gator (1) a s s i s t s the phonon p r o p a g a t o r in enhancing the e l e c t r o n m a s s as m* = m(1 + X + ~), where X is the d i m e n s i o n l e s s p a r a m e t e r of e l e c t r o n - p h o non coupling. The m a s s enhancement due to the spin has been pointed out r e c e n t l y by Kondo [4]. In a superconducting alloy, the spin p r o p a g a t o r a p p e a r s with the " r e p u l s i v e " sign in the i n t e g r a l equation to d e t e r m i n e the energy gap A(c~) as the function of the e l e c t r o n excitation energy ~. This is because the exchange s c a t t e r i n g tends to hinder the f o r m a t i o n of Cooper p a i r s . But whether it is r e a l l y r e p u l s i v e depends on the r e l a t i v e magnitude of the Zeeman energy a)2 in c o m p a r i s o n with the

108

15 August 1965

energy gap; the Cooper p a i r is affected by the r e p u l s i v e p a r t of (1) at low e n e r g i e s only when 2 is l a r g e r than the energy gap. Note that the c h a r a c t e r i s t i c phonon energy a~1 is usually much higher than co2 and also that the phonon induced a t t r a c t i o n is p r a c t i c a l l y constant over the wide range 0 < Icol
~(~) =~(I + ~ u(~)).

(2)

Here A is the gap in the phonon dominating region ~2 << l~l <~~1, and ~ indicates the degree of the effect of the exchange coupling. W h e n A << °°2, these are given by

= ~ o [ 1 - ~b ~'] ,

~ = ~b

(3)

where A o is the gap in the pure metal and b = log (2~2~Ao). Our model suggests that the gap p a r a m e t e r A(c~) will exhibit an anomalous dispersion at co = c~2 as indicated by (2). If this is the case, we should be able to detect the dispersion by the use of tunneling just in the same way as we observe the phonon spectrum superimposed on the B C S density of states. The detail of derivation of above results will be published later. I wish gratefully to acknowledge the kind hosp t t s l i t y of P r o f e s s o r Sir NeviU Mort and also d i s cussions with Dr. K. H. Bennemann.

References 1. A.A. Abrikosov and L. P. GorWkov, Soviet Phys. JETP 12 (1961) 1243. 2. G.M. Eliashberg, Soviet Phys. JETP 11 (1960) 696. 3. A.B.Mtgdal, Soviet Phlm.JETP 7 (1958) 996. 4. J.Kondo, Progr.Theor. Phys. 33 (196,5) 575. 5. P. Morel and P. W. Anderson, Phys. Rev. 125 (1962) 1263.