CURRENT-PHASE RELATION OF SNS AND SS1S WEAK LINKS

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CURRENT-PHASE RELATION OF SNS AND SS1S

WEAK LINKS

H. Fink, R. Poulsen

To cite this version:

JOURNAL DE PHYSIQUE _{Colloque C6, suppl&tnent au no 8, Tome 39, aoat 1978, page C6-558}

CURRENT-PHASE RELATION

OF

S N S

AND

SS,S

WEAK

LINKS

H.J. Fink and R.S. Poulsen

Department of EZectricaZ Engineering ~ n i v e r s i t y o f CaZifornia, Dauis, CA 95616 U.S.A.

R6sumd.- Des solutions exactes pour la relation phase-courant et les courants critiques de "weak links" SS,S et SNS sont calculdes pour differentes conditions aux limites et les longueurs des r6-

gions S, ou N. Quand les longueurs de S, ou N, soient 2d

,

sont com arables 1

lcnl

ou infGrieures,

les deux types de jonctions ont le msme comportement.

Q U L ~

2dn >> T61, une jonction SNS se com-

porte comme une jonction tunnel Josephson, mais une jonction SSIS comme une spire supraconductrice,

avec seulement quelques aspects des effets Josephson.

Abstract.- Exact solutions of the current-phase relations and the critical currents of SS,S and SNS weak links are calculated for different boundary conditions and lengths of the S, and N regions.

When the lengths of the S, and N re ions, 2d

,

are comparable to

lcnl

or smaller, both types of weak

links behave similarly. When 2d >> fcnl, a

SPS

weak link behaves very much like a Josephson junction

n

but a SS,S weak link like a long superconducting wire with some of the Josephson-like properties re- maining.

WEAK LINKS.- A weak link is a region which is in intimate contact with and located between two stron- gly superconducting regions and is able to transmit some of the long range superconducting properties between the strongly superconducting regions.

The phase difference across the center region of a SS S weak link which is a superconductor In

the superconducting state (T < T bulk of S,) is

proportional to its length and current density. These solutions /1,2/ give rise to a sinusoidal- like current-phase relationship for a short link which is fortuitously similar to that of a Joseph- son Junction.

The SNS weak link center region is not in the superconducting state when not in contact with the outside superconductors. The center region aquires superconducting properties when sandwiched between the outside superconductors due to the proximity effect. The maximum loss free dc current is then the critical current of the SNS weak link.

The essential difference between SS S and SNS weak links is that the phase difference sustai- ned over the center region of a SNS weak link ap-

proaches the value ~ / 2 at the critical current den-

sity when 2dn/1

5,1 >>

1, whereas for a long SS,S weak link it is proportional to the length of cen-

ter region 2d

.

n

THEORY.- An SNS weak link was treated analytically

in /3/ and the results were compared with experi-

ments by Clarke 141. Here we present exact numeri-

cal solutions for various material parameters and

boundary conditions for both SNS and SSIS weak

links. Definitions for the purpose of normalization are in /3/. Boundary conditions for a continuous

current flow across the NS boundaries are /3/

XGs and X are the Gor'kov parameters /5/ which

Gn

are functions of temperature T and

II

and II

n7

pectively. The parameter a determines whether the

weak link is SS,S or SNS. If a > 0, then the center

region of the weak link is a superconductor below

the S, bulk transition temperature. For a < 0 the

.renter region is denoted by N.

SNS WEAK LINK.- Since our SNS structure is assumed to be symmetric, it is only necessary to find solu-

tions between x = 0 and x = x,. We assume that x

= 10, unless otherwise specified, so that the S-

regions approach the bulk limit (x, = 105 in con-

ventional units). The value x is varied. The thick ness of the N-region, 2dn, is therefore varied. We find that there are at least two simple solutions of f and y for a fixed value of x and current den- sity i which satisfy the boulidary conditions eqs.

(1) and (2) and also df/dx = 0 at x = 0 and dy/dx

= O a t x = x .

2

The total phase difference of the superconduc-

ting phase across the SNS or SS S sandwich is

where the second term on the right-hand side is the

phase difference 2 ( ~ -XI) across the N or S1 re-

gion.

The phase differences across the N-region on- ly, which are very similar to the approximate resu- lts 131, approach those of the Josephson dc current phase relation in the very weak coupling limit

(2d >> I) 161. This is not the case for a SS S n

weak link.

Figure 1 shows the total phase difference

-2(x2-xO) across a SNS weak link as a function of current density i, calculated from eq. (3).

14

.

I 1 I

EQ.

Fig. 1 : Current density-phase relation across a

SNS (a < 0) weak link with x l = 10 for various va-

lues of x2. The envelope, Eq., is that for an in-

finitely long wire calculated over a length x, = x,

= 10 (no N-region). A solution for a short weak

link xl = 2 and x, = 2.5 is also shown.

The envelope is calculated from i = Q(I-Q') with Q

=

-(x

-

x

) 110. The latter corresponds to the dc

2 0

current-superfluid velocity relation for an infini- tely long, uniform wire. The curve denoted by Eq.

in figure 1 is calculated with a N-region of zero

length. The lower branch splits up into two solu- tions when a very small N-region is inserted into a long superconducting wire.

SS S WEAK LINK- It is found that the critical cur-

rent -2(x

-X

) , the phase difference across the S

2 1

region of a weak link, is smaller than n/2 if S is very thin and larger than n/2 if S is thick.

Figure 2 shows -2(x

-X

) the total phase difference

2 0

across the SSIS weak link, for various thicknesses

of the S1 region. The curve Eq. corresponds to the

uniform solution with no S region present. The differences between SS S and SNS weak links become

very much apparent when comparing figure I and 2.

Fig. 2 : Supercurrent-phase relation across a SSIS

(a > 0) weak link (from x = 0 to 2x2) for various values of x,. The curve denoted by eq. corresponds to the uniform solutions of a very long wire calcu-

lated over a length x2 = x l = 10 (no S1 region).

Figure 3 summarizes the results for ic, the GL normalized critical current density, for various

parameters y and a as a function of xz all with

b = 0.45 and xl = 10. The curves for a<0 are the

results of SNS weak links, and for a>0 those of

SSlS weak links.

Fig. 3 : Comparison of the critical current densi-

ties ic of SSlS (a > 0) and SNS (a < 0) weak links as a function of the thickness of the Si and N

regions all for xi = 10and ms/mn = 2 for various

CONCLUSIONS.- It i s a p p a r e n t t h a t a SNS weak l i n k i s a k i n t o a J o s e p h s o n j u n c t i o n , i n p a r t i c u l a r i n t h e weak c o u p l i n g l i m i t ((x2-x ) >> I ) , and t h a t a SS,S weak l i n k w i t h a long S l r e g i o n assumes proper- t i e s which c o r r e s p o n d m a i n l y t o t h o s e o f a n i n f i n i - t e l y l o n g s u p e r c o n d u c t i n g w i r e b u t i n c l u d e some J o s e ~ h s o n - l i k e b e h a v i o r 171. When t h e N o r S1 r e - g i o n s a r e comparable i n l e n g t h t o

1 ~ ~ 1 ,

t h e b u l k c o h e r e n c e l e n g t h of N o r S, r e g i o n , t h e long r a n g e c o h e r e n c e p r o p e r t i e s of SNS and SSIS weak l i n k s a r e v e r y s i m i l a r .

One r e m a i n i n g problem c o n c e r n s t h e e x a c t boundary c o n d i t i o n s o f t h e p a i r p o t e n t i a l s a t t h e SN i n t e r f a c e . It was found / 3 / t h a t B

"

0 . 3 a t t h e NS boundary f o r Pb-Cu. D e u t s c h e r e t a l . 181 found t h a t t h e p r o b a b i l i t y f o r e l e c t r o n s t o p e n e t r a t e t h e SN b a r r i e r 0 = 0.3 f o r Pb-Cu. S i m i l a r r e s u l t s were

found by Adkins and Kington 191 (0

"

0.25) and

F r e a k e / l o / ( 0 = 0.29). It i s p o s s i b l e t o r e l a t e B t o 0 . T r e a t i n g AG a s a t r a v e l i n g wave i n c i d e n t on t h e SN boundary from t h e S r e g i o n we g e t 0 = (2B/ ( I + B ~ ) ) ~ . With B = 0.30 t h e v a l u e of 0

"

0.30 i n good agreement w i t h r e f e r e n c e s / 8 , 9 and 101.

R e f e r e n c e s

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