THE JOSEPHSON EFFECT IN SUPERCONDUCTING CONSTRICTIONS : MICROSCOPIC THEORY

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THE JOSEPHSON EFFECT IN

SUPERCONDUCTING CONSTRICTIONS :

MICROSCOPIC THEORY

I. Kulik, A. Omelyanchouk

To cite this version:

JOURNAL DE PHYSIQUE Colloque C6, suppliment au no

8,

Tome 39, aoiir

1978,

page

_{~ 6 - 5 4 6}

THE JOSEPHSON EFFECT IN SUPERCONDUCTING CONSTRICTIONS

:

MICROSCOPIC THEORY

1.0. Kulik and A.N. Omelyanchouk,

Phgsico-TechnicaZ I n s t i t u t e of

Low

Temperatures of the UkrSSR Academy

of

Sciences, Lenin's av.

47

Kharkov

310164,

U.S. S. R.

R6sumb.- L ' e f f e t Josephson c o n t i n u dans des microponts supraconducteurs e s t btudid dans l e c a d r e de l a the'orie microscopique de l a s u p r a c o n d u c t i v i t b . Les r e l a t i o n s c o u r a n t - p h a s e I ( @ ) e t l a grandeur du courant c r i t i q u e I c ( T ) obtenues d i f f s r e n t des grandeurs correspondantes pour l e s j o n c t i o n s t u n n e l s . La d i s t i n c t i o n e s t l a p l u s marqu6e aux b a s s e s temperatures e t pour l a l i m i t e pure @ > > a ) , P, : l i b r e parcours moyen des d l e c t r o n s , a : rayon d e l ' o r i f i c e . Dans l a l i m i t e p u r e (P.>>a) l a r e l a t i o n c o u r a n t phase e s t l a s u i v a n t e

T A ~ ( T ) Ao(T)cos@/2

I ( $ ) =

-

'RN

s i n + l 2 tanh 2T

A b s t r a c t . - The d.c. Josephson e f f e c t i n superconducting microbridges h a s been s t u d i e d i n terms of t h e microscopic theory of s u p e r c o n d u c t i v i t y . Current-phase r e l a t i o n , I ( $ ) , and a c r i t i c a l c u r r e n t v a l u e , I c ( T ) , have been o b t a i n e d and shown t o be d i f f e r e n t from those q u a n t i t i e s corresponding t o t h e tunnel j u n c t i o n . These d i f f e r e n c e s a r e most pronounced a t low temperatures and f o r t h e pure c o n s t r i c t i o n s : Il>>a, II : being t h e e l e c t r o n mean f r e e p a t h and a : t h e r a d i u s of t h e o r i f i c e con- n e c t i n g two superconductors. I n t h e l a t t e r c a s e

nAo(T) AO(T) cos@/2

I ( @ ) =

-

e%

s i n 412 tanh 2T

T i l l r e c e n t l y , t h e Josephson e f f e c t i n super- I ( @ ) and I (T), a r e d i f f e r e n t from t h e corresponding conducting microbridges has been considered i n dependences f o r t h e tunnel j u n c t i o n . These d i f f e - terms of the Ginsburg-Landau t h e o r y v a l i d f o r tem- r e n c e s a r e most pronounced a t low temperatures and p e r a t u r e s c l o s e t o T

.

The t h u s o b t a i n e d r e s u l t s f o r extremely pure microbridges.

c o i n c i d e w i t h those f o r tunnel j u n c t i o n s ( i n t h e The current-phase r e l a t i o n i n t h e l i m i t a<<II, proper temperature range). Here we a r e going t o a < < t 0 / 3 / does n o t depend on t h e r e l a t i o n between

5"

perform a microscopic a n a l y s i s of c u r r e n t - c a r r y i n g and

R

and i s expressed through t h e j u n c t i o n r e s i s - s t a t e s i n microbridges v a l i d f o r a r b i t r a r y tempera- t a n c e i n t h e normal s t a t e , %

t u r e s . A ~ < T ) c o s ~ 1 2

The problem i s e s s e n t i a l l y space-inhomogene- I ( @ ) = (1)

2T ous and i s t r e a t e d i n terms of t h e q u a s i c l a s s i c a l

'Yhe v a l u e of %is determined by t h e r e l a t i o n / 4 / s u p e r c o n d u c t i v i t y theory e q u a t i o n s , t h e Eilenber- _{e 2 s . s F}

g e r e q u a t i o n s / l / . The b r i d g e i s modelled a s an

s1

2 (2aM) _. . . * ' ( 2 )

o r i f i c e of r a d i u s a i n the n o n t r a n s p a r e n t s h e e t se-

parating two superconducting semispaces. The situa- S being t h e c r o s s - s e c t i o n of t h e o r i f i c e ,

SF

t h e t i o n i s e s s e n t i a l l y d i f f e r e n t i n t h e d i r t y l i m i t , Fermi s u r f a c e a r e a . A t T = 0 , t h e phase dependence %>>a, and i n t h e pure l i m i t , L>>a. I n t h e former of t h e c u r r e n t f o r t h e pure m i c r o c o n s t r i c t i o n expe-

r i e n c e s jumps a t p o i n t s @ = + n ( f i g u r e l ) . The c r i t i - c a s e , t h e E i l e n b e r g e r e q u a t i o n s a r e reduced t o d i f -

c a l c u r r e n t of t h e c o n s t r i c t i o n , Ic, exceeds t h e f u s i o n type ones, t h e Usadel e q u a t i o n s , and t h e i r

solution is essentially isotropic /21.

In

the lat- t u n n e l j u n c t i o n c r i t i c a l c u r r e n t f o r t h e e q u a l nor- t e r c a s e , t h e s o l u t i o n f o r t h e Green f u n c t i o n s h a s mal-state a t =

'

a "stream-like" c h a r a c t e r , i . e . s h a r p l y a n i s o t r o p i c

_-

'c 1.32 f o r P . < < a

i n t h e momentum space 131. I 0 2 f o r % > > a (3)

I n t h e l i m i t a<<P,, a<<SO

(eO

i s t h e coheren- _{The o r d e r parameter modulus a t t h e p l a n e of} c e l e n g t h ) , t h e e x a c t s o l u t i o n o f t h e E i l e n b e r g e r the ..ifice ( z = 0 ) equals

e q u a t i o n s has been o b t a i n e d by i n t e g r a t i n g a l o n g _A(0) = AOcos$/2 ( 4 ) t h e " t r a j e c t o r i e s " which a r e t h e c h a r a c t e r i s t i c s and a t T = 0 f o r t h e pure p o i n t - c o n t r a c t j u n c t i o n , of t h e e q u a t i o n s . The r e s u l t s o b t a i n e d s u g g e s t

t h a t t h e f u n c t i o n s mentioned i n t h e a b s t r a c t ,

A(0)

=

0 for the current equal to Ic. The plot of

+

the surfaces lA(r)

I

=

const. is shown schematically

References

at figure

2.

/l/ Eilenberger, G., 2s. Phys.

214

(1968)195.

Fig.

1 :

Current-phase relations in the pure limit

for different temperatures.

( 1 )

T

= 0 ;

(2) TIT

=

0.2

;

(3) TITc

=

0.98. Io(T)

rA9(T)

=-

A~

($1

2 e R ~

tanh

-

2T

The "healing length" of the order parameter equals

to

'L

a, and its space dependence has the f q m

+

i. signz. sin@/2

(

I

-

'

$

)l

,(5)

while Green functions take their equilibrium values

at the superconducting "banks" at distances

r -5

,

where

1

-

1 1

5

-

q+-ji;.

-t

In

( 5 ) ,

'

(

r

)

is the solid angle at which the

+

orifice is seen at the point r. Eq.(I)relates

to the

junction of two identical superconductors :Ao

=

A

(T)

=

A2(T), A

-

the equilibrium values of the energy

1 , 2

gaps

.

Fig.

2 :

The cross-section of the superconducting

point-contact junction (microconstriction) showing

lines of constpt value of the modulus of the order

parameter, lA(r)

l

=

const. at

@=IT.

121 Kulik, I.O., Omelyanchouk, A.N., Proceedings of

LT-14,

5

(1975)

116 ;

Zh.Exp.Th.Phys.Lett.

2

(1975) 216.

/ 3 /

Kulik, I.O., Omelyanchouk, A.N., Fiz.Nizk.Temp.

(Low Temp.Phys., Kharkov)

2

(1977) 945

;

5

(1978) No 4.

141 Kulik, I.O., Shekhter, R.I., Omelyanchouk, A.N.,

Solid State Comun.

2

(1977) 301.

For the asymmetrical junction Al<<A2,

the current-

phase relation can be shown to be sinusoidal in

@