On the transition temperatures of granular superconductors

V o l u m e 31A, n u m b e r 1

ON T H E

PHYSICS

TRANSITION

TEMPERATURES

LETTERS

OF

12 J a n u a r y 1970

GRANULAR

SUPERCONDUCTORS

H. Bi)TTNER and E. G E R L A C H

Battelle-lnstitut e. V., Frankfurt~Main, Germany Received 29 November 1969

It is shown that Friedel oscillations in granular superconductors give rise to an energy gap at the Fermi surface which reduces the transition temperature.

Recently, the t r a n s i t i o n t e m p e r a t u r e s T c of s o m e g r a n u l a r s u p e r c o n d u c t o r s , e.g. ~ - L a [1], have been o b s e r v e d to d e c r e a s e with d i m i n i s h i n g g r a i n size. T h e s e r e s u l t s a p p a r e n t l y cannot be explained by a change of the phonon s p e c t r u m [2] or of the d e n s i t y of s t a t e s , as d i s c u s s e d by P a r m e n t e r , which lead to an i n c r e a s e of the bulk t r a n s i t i o n t e m p e r a t u r e T c. As a p o s s i b l e e x p l a n ation of this d e c r e a s e Strongin [4] c o n s i d e r e d the fluctuations of the o r d e r p a r a m e t e r . We p r o p o s e an additional m e c h a n i s m , which gives r i s e to d e c r e a s e d t r a n s i t i o n t e m p e r a t u r e s T c. This is b a s e d on the e x i s t e n c e of an e n e r g y gap at the F e r m i s u r f a c e due to F r i e d e l o s c i l l a t i o n of the c h a r g e density at the g r a i n s u r f a c e s . The o c c u r r e n c e of a s i m i l a r type of e n e r g y gap o r i g i n a t i n g f r o m the s c r e e n i n g of e x t e r n a l e l e c t r i c a l fields was d e s c r i b e d by B a l k a r e i and S a n d o m i r s k i i [5]. In o r d e r to e s t i m a t e the F r i e d e l o s c i l l a t i o n s in a s m a l l cubic g r a i n volume L 3, we c o n s i d e r a s i m p l i f i e d model. We c a l c u l a t e the o s c i l l a t i o n s for a m e t a l l i c half space and u s e this r e s u l t for all s u r f a c e s of the g r a i n . S t a r t i n g f r o m the e l e c t r o n wave function ~P = (2/L3) ½ sinkzz e x p i

(kxx+kyy)

(1)

one obtains the c h a r g e d e n s i t y (~ = 2kFZ )

p = (ek3F/lr2~ 3) (sin ~

- ~ cos ~)

(2)

which in t u r n , is r e l a t e d by P o i s s o n ' s f o r m u l a to the potential e n e r g y

U = (e2kF/8Eo)(sin

~/~ + ~ Si ~ + cos ~ - ½~). (3)

In the l i m i t of l a r g e v a l u e s of ~ we obtain

U~ = (e2kF/4Eo) cos ~/~2

.

f.0

~/r¢o Q$,

& ~5 m Fig. 1. Transition temperature as a function of 5.

(e2/4~okF L2) and of f r e q u e n c y (~/kF) gives r i s e to an e n e r g y gap of width 5

= e2/2CokF L2

(5)

c e n t e r e d at the F e r m i e n e r g y . In our model we a s s u m e that 5, as given by eq. (5), is a l s o v a l i d for the t h r e e - d i m e n s i o n a l g r a i n . On account of these o s c i l l a t i o n s the m e t a l l i c state changes into a s e m i c o n d u c U n g state with a 4 o n a r r o w ga~ of the o r d e r ofol0- eV for L ~ 20 A and of 10-o eV for L ~ 60 A. If 5 b e c o m e s c o m p a r a b l e with the s u p e r c o n d u c t i n g gap Ao a d r a s tic d e c r e a s e of Tc, o i s to be expected. The t h e o r y of the t r a n s i t i o n t e m p e r a t u r e of a s u p e r conducting s e m i c o n d u c t o r was d e r i v e d by Little [6], which in our case i m m e d i a t e l y l e a d s to the r e l a t i o n between (Tc/T c o) and (5/2A o) plotted in fig. 1. In addition, of ~ourse, t h e r e a r e the c o n t r i b u t i o n s by the other effects m e n t i o n e d above.

(4) 1

F o r two p a r a l l e l s u r f a c e s at d i s t a n c e L >> ~k F 1 equation (4) leads to a r a n g e of length ~L around the c e n t e r where the a m p l i t u d e of U~ i s n e a r l y constant. This p e r i o d i c potential of a m p l i t u d e

References 1. J.J. Hanak and J.I. Gittleman, Int. Conf. on the Science of superconductivity, August 1969, paper 9.1.

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