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APPLICABILITY OF THE THEORY OF THE
JOSEPHSON EFFECT IN TUNNEL JUNCTIONS TO
WEAK LINKS
A. Zorin, K. Likharev
To cite this version:
JOURNAL DE PHYSIQUE
Colloqur C6, supplPmenr
au
no8 , Tome 39, aotit 1978, page
~ 6 - 5 7 3
A P P L I C A B I L I T Y OF THE THEORY OF THE JOSEPHSON EFFECT I N TUNNEL JUNCTIONS TO WEAK L I N K S
A . B . Zorin and K.K. Likharev
Department of Physics, Moscow State University, Moscow 227234
(U.S.S.R.)
Rdsum6.- On examine un modsle de j o n c t i o n Josephson 02 l a b a r r i g r e q u i s d p a r e l e s supracon- d u c t e u r s c o n t i e n t un grand nombre de p e t i t s o r i f i c e s ( a s k - ] ) . On montre que pour c e t t e F s t r u c t u r e l ' e f f e t Josephson s e d g c r i t 2 l ' a i d e de l a t h d o r i e b i e n connue ddveloppde pour l e s jonc t i o n s t u n n e l s .
A b s t r a c t . - A model of weak l i n k i s c o n s i d e r e d , where superconductors a r e s e p a r a t e d by t h e opaque b a r r i e r w i t h a g r e a t number of s m a l l ( a
s
k -1) t r a n s p a r e n t a p e r t u r e s i n i t . The Josephson e f f e c t i n t h e s t r u c t u r e is shown t o be a!equately d e s c r i b e d by t h e well-known t h e o r y f i r s t suggested f o r tunnel j u n c t i o n s .INTRODUCTION.- The problem of a p p l i c a b i l i t y of t h e well-known theory of t h e Josephson e f f e c t i n t u n n e l
j u n c t i o n s / 1 , 2 / t o weak l i n k s was d i s c u s s e d many ti- mes a f t e r t h e r e p o r t by Zimmerman / 3 / . I f a weak l i n k , i . e . t h e Josephson s t r u c t u r e w i t h d i r e c t (non- tunnel-type) c o n d u c t i v i t y between superconducting e l e c t r o d e s , h a s a l a r g e s i z e a 25 i t s p r o p e r t i e s a r e c e r t a i n l y d i f f e r e n t from t h o s e of a t u n n e l j u n c t i o n / 4 / . However, i n t h e c a s e of a r e l a t i v e l y s m a l l weak l i n k ( a 55) t h i s d i f f e r e n c e i s n o t s o e v i d e n t . I n f a c t , t h e I-V curves of t h e Josephson j u n c t i o n s w i t h r e l a t i v e l y s m a l l i n t r i n s i c c a p a c i - tance ( B =
~ ~ I ~ G c / M
5 I ) , c a l c u l a t e d / 5 , 6 / from t h e " t u n n e l theory", appear t o b e c l o s e to t h e experimental I-V c u r v e s of some small weak l i n k s .Recently, Kulik and Omelyanchuk have shown t h a t t h e dc Josephson e f f e c t i n small (a <<g) weak l i n k s i s s i g n i f i c a n t l y d i f f e r e n t from t h a t i n t u n n e l j u n c t i o n s f o r b o t h small (!,<<a) / 7 / and l a r g e (R>>a)
/ 8 / v a l u e s of t h e e l e c t r o n mean f r e e p a t h i . So t h e problem a r i s e s a s t o whether t h e r e r e a l l y a r e weak l i n k s f o r which t h e tunnel theory i s v a l i d . I n t h i s paper we show t h a t t h i s t h e o r y i s a p p l i c a b l e t o a t l e a s t one weak l i n k model.
I n t h i s model, two superconducting e l e c t r o - d e s a r e assumed t o be s e p a r a t e d by a t h i n i n t e r l a y e r which i s n o n t r a n s p a r e n t t o e l e c t r o n s everywhere w i t h t h e e x c e p t i o n of a l a r g e number of v e r y small aper- t u r e s . These a p e r t u r e s ("holes") form m i c r o s h o r t s w i t h d i r e c t c o n d u c t i v i t y p r o v i d i n g weak l i n k a g e of
t h e l a s t c o n d i t i o n (more e x a c t l y , k a F
s
1) can be f u l f i l l e d only f o r t h e h o l e s of atomic o r d e r . However, i n semimetals and d e g e n e r a t e semiconduc-t o r s t h e h o l e s can be much l a r g e r than t h e atomic s i z e , and t h e medium i n s i d e t h e m i c r o s h o r t s can be t r e a t e d a s a continuum. For a n a l y s i s of t h e model
we have used t h e method s i m i l a r t o t h a t d i s c u s s e d i n t h e paper
191,
where i t was a p p l i e d t o t h e tun- n e l j u n c t i o n s w i t h inhomogeneous b a r r i e r s .SINGLE APERTURE.- We s t a r t w i t h t h e c a l c u l a t i o n of wave f u n c t i o n s f o r a s i n g l e round h o l e i n s i d e t h e opaque b a r r i e r . The f u n c t i o n s a r e found a s solu-
t i o n s of t h e SchrGdinger e q u a t i o n , which f o r our c a s e reduces t o t h e wave e q u a t i o n
v 2 Y + k 2 Y = 0 , k 2 = 2 m E / M ( 1 ) everywhere w i t h t h e e x c e p t i o n of t h e n o n t r a n s p a r e n t p a r t of t h e i n t e r l a y e r . Equation (1) w i t h a p p r o p r i a t e boundary c o n d i t i o n s i s solved under t h e assump- t i o n t h a t t h e p l a n e wave
+
+Yo = e x p ( i k o r ) (2) comes t o t h e b a r r i e r p l a n e (x=O) from t h e f i r s t superconductor (x < 0 ) . The wave f u n c t i o n i n t h e o t h e r e l e c t r o d e ( x 0) is c a l c u l a t e d u s i n g t h e p e r t u r b a t i o n t h e o r y w i t h r e s p e c t t o t h e small r a - d i u s a of t h e h o l e . The s o l u t i o n of t h i s d i f r a c t i o n problem i n t h e r e g i o n x > 0, r >> a a p p e a r s a s
+
kkoxa+ +
Y ( r ) =- -
3a ' X P ~ ' ~ ) c o s ( n , r ) ,I Z I
=1 ~ ~ 1
(3)+
The f u n c t i o n Y ( r ) d e s c r i b e s a s p h e r i c a l t h e e l e c t r o d e s . wave t r a n s m i t t e d through t h e b a r r i e r w i t h t h e am-The s i z e o f t h e h o l e s a r e considered t o be p l i t u d e f a l l i n g o f f t o z e r o a s soon a s t h e d i r e c U
+
small compared n o t o n l y w i t h R and 5 b u t a l s o w i t h t i o n of wave v e c t o r k approaches t h e p l a n e of b e t h e Fermi wavelength : $a << 1 . I n t h e u s u a l m e t a l s b a r r i e r . The amplitude a n g u l a r d i s t r i b u t i o n -Is
+
i n d e p e n d e n t o f t h e d i r e c t i o n ko o f t h e i n c i d e n t
+
wave ( 2 ) ; ko a f f e c t s o n l y t h e t o t a l i n t e n s i t y of
+ +
t h e t r a n s m i t t e d wave :
I Y I
= cos ( k o , n ) . The ampli- t u d e i s a l s o p r o p o r t i o n a l t o t h e h o l e s i z e cubed, t h e l a s t r e s u l t b e i n g v a l i d f o r t h e a p e r t u r e s o f a n y s h a p e . MULTITUDE OF APERTURES.- I n t h e c a s e o f a l a r g e number o f s m a l l h o l e s l o c a t e d i n t h e b a r r i e r w i t h -2 n o t v e r y h i g h s u r f a c e d e n s i t y n L a,
t h e t r a n s - m i t t e d wave c a n b e c a l c u l a t e d a s a l i n e a r superpo- s i t i o n o f s i n g l e h o l e f u n c t i o n s ( 3 ) . A t d i s t a n c e s x >> k-' i t g i v e s t h e f o l l o w i n g r e s u l t : 2+
-+Y
( r ) =-
7
i ( n a 2 ) ( k o x a ) e x p ( i k o r ) , a = < a ? >113
(4) Comparison of e q u a t i o n s (2) and ( 4 ) shows t h a t t h e t r a n s m i t t e d wave p a s s e s through t h e b a r r i e r w i t h o u t r e f r a c t i o n . E x p r e s s i o n ( 4 ) i s e x a c t l y t h e same a s f o r a homogeneous t u n n e l j u n c t i o n w i t h t h e b a r r i e r t r a n s p a r e n c y and, c o n s e q u e n t l y , w i t h t h e normal r e s i s t a n c e p e r u n i t a r e a-
2 R = k p 36112 ( n a 2 ) - 2 ( k F a ) - 2 ~ q , ( 6 ) where R = M/2e2 %2k 0 i s a quantum u n i t o f r e s i s -9 t a n c e
.
The f a c t o f e q u a l i t y o f wave f u n c t i o n s f o r o u r weak l i n k model and f o r t h e t u n n e l j u n c t i o n shows t h a t a t a l l d i s t a n c e s e s s e n t i a l f o r supercon-
a p p r o x i m a t e l l y v a l i d n o t o n l y a t kFa < < I b u t a l s o a t kFa s 1 , we c a n t a k e kF-l= a
=
1 0 - ~ c m , n - l l 2 = 3 a and g e t t h e e s t i m a t e6
% 0 . 2 . This v a l u e o f 6 cor- r e s p o n d s t o a n e g l i g i b l y s m a l l e f f e c t of c a p a c i - t a n c e on t h e I-V c u r v e of t h e JosephBon j u n c t i o n151.
CONCLUSION.- At l e a s t one model of t h e Josephson weak l i n k w i t h d i r e c t c o n d u c t i v i t y c a n f o l l o w t h e
t u n n e l t h e o r y . The reduced c a p a c i t a n c e o f t h a t s t r u c t u r e c a n b e l e s s t h a n u n i t y , s o i t s I-V c u r v e c a n b e c l o s e t o t h o s e o b s e r v e d e x p e r i m e n t a l l y i n weak l i n k s . The model c a n p o s s i b l y d e s c r i b e some o f t h e p o i n t c o n t a c t j u n c t i o n s .
ACKNOWLEDGMENTS.- H e l p f u l d i s c u s s i o n s w i t h 1.0. Kulik and A.N.Ornelyanchuk a r e h i g h l y a p p r e c i a t e d .
R e f e r e n c e s
/ I / Werthamer, N.R., Phys. Rev.
147
(1966) 225. / 2 / L a r k i n , A . I . and Ovchinnikov, Yu.N. Sov. Phys.JETP
3
(1967) 1035./3/ Zimrnerman, J . E . P r o c . Appl. Supercond. Conf. (Annapolis) IEEE P u b l . 72-CH0682-TABSC, p.544. / 4 / L i k h a r e v , K.K., S u p e r c o n d u c t i n g Weak L i n d s , t o
b e p u b l i s h e d .
/ 5 / McDonald, D.G., Johnson, E.G. and H a r r i s , R.E. Phys. Rev.
B13
(1976) 1028.161 Z o r i n , A.B. and L i k h a r e v , K.K. F i z i k a Nizk.Ternp. 3 (1977) 148 (Engl.Trans1. : Sov. J.Low Temp.
-
Phys .)
.
d u c t i v i t y (% 5 ,!?. ) t h o s e two s t r u c t u r e s a r e c o m p l e / 7 / Kulik, 1 . 0 . and Omelyanchuk, A.N. JETP L e t t . - 2 1 (1975) 96. t e l y i d e n t i c a l . Thus, we may n o t complete t h e u s u a l / 8 / Kulik,I.O. and Omelyranchuk, A.N. F i z i k a Nizk. c a l c u l a t i o n s of t h e c u r r e n t between e l e c t r o d e s and Temp.
2
(1977) 945 ( W l . T r a n s l . : Sov. J.Low may u s e t h e r e s b l t s o f t u n n e l t h e o r y ( f o r b o t h t h e Temp. Phys .).
dc and ac Josephson e f f e c t s ) , taking expression ( 6 ) 191 K u l i k , I . O . , M i t z a i , Yu.N. and h e l ~ a n c h u k , A . N . Zh.Eksp. T e o r . F i z .
66
(1974) 1051 (Engl. f o r t h e normal r e s i s t a n c e.
T r a n s l . : Sov. Phys. JETP).CAPACITANCE.- E x p e r i m e n t a l weak l i n k s u s u a l l y have v e r y small c a p a c i t a n c e
