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On the thermodynamics of weakly connected superconducting rings
Volume 26A. n u m b e r 3
PHYSICS
1. I . M . K h a l a t n i k o v . Z h . E k s p . i T e o r F i z . 22 (1952) 687. 2. L . J . C h a l l i s , P r o c . P h y s . Soc. 80 (1962) 759. 3. K u a n g W e y - Y e n . Zh. E k s p . i T e o r . F i z . 42 (1962) 921; Soviet P h y s . J E T P 15 (1962) 635. 4. R . C . J o h n s o n a n d W . A . L i t t l e , P h y s , R e v . 130 {1963) 596. 5. D . A . N e e p e r a n d J . R . D i l l i n g e r , P h y s . R e v . 135 (1964) A1028,
ON
THE
LETTERS
1 J a n u a r y 1968
6. L . J . C h a l l i s a n d J . D , N . Cheeke. P r o g r e s s in r e f r i geration science and technology {Pergamon P r e s s . for the International Institute of Refrigeration) 1 (1965) 227, 7, D . A . N e e p e r , D . C . P e a r c e a n d R . M . W a s i l i k , Phys. Rev. 156 (1967) 764. 8. J . I . G i t t l e m a n and S.Bozu~,ski. Phys. Rev. 128 {1962) 646, 9. R . C . J o h n s o n , Bull. Am, Phys. Soc. 9 {1964) 713.
THERMODYNAMICS OF SUPERCONDUCTING
"WEAKLY RINGS
CONNECTED
S. N A K A J I M A a n d Y. K U R O D A * Institute for Solid State Physics, University of Tokyo, Minato-ku, Tokyo
Received 1 D e c e m b e r 1967
A t h e r m o d y n a m i c a l analysis is made of the magnetization of weakly connected superconducting rings as o b s e r v e d by Silver and Z i m m e r m a n . A possible decay of the p e r s i s t e n t c u r r e n t is predicted.
W h e n a w e a k l i n k (point c o n t a c t o r t u n n e l j u n c tion) is introduced into a superconducting ring [1], t h e j u m p in t h e p h a s e of t h e o r d e r p a r a m eter across the link is no longer an integral mult i p l e of 2~. If t h e r i n g i s t h i c k c o m p a r e d w i t h t h e penetration depth and the link is so small that the magnetic flux through it may be ignored, the p h a s e j u m p i s g i v e n by ( 2 1 r ~ / ~ o ) , w h e r e 4~ i s t h e f l u x e n c l o s e d b y t h e r i n g a n d ~ o = (~/~c/e). In t h e p r e s e n c e of t h e e x t e r n a l f l u x 4)x, t h e G i b b s p o t e n t i a l of o u r s y s t e m t a k e s t h e f o r m
3 0-9~. 20
(a)
10 00 3 50 05
20 A
G : GO + (1/2L)(~-4)x)2
- E o c o s ( 2 ~ / 4 ) o)
(1)
(b)
Here Go is a 4)-independent part, L is the induct a n c e of t h e b u l k r i n g , a n d t h e l a s t t e r m i s t h e p h a s e c o u p l i n g e n e r g y w h i c h r e s u l t s in t h e J o s e p h s o n c u r r e n t [2] a c r o s s t h e l i n k . In t h e s t a t i c l i m i t , tim s y s t e m m u s t b e i n t h e t h e r m o dynamic state for which G is a minimum with r e s p e c t to ~): sin[2~/4~o]
= - 27rot[((b/~o) - ( 4 ) x / 4 ) o ) ]
10
oo
106
-0.96
Io
oo ~" o'~/ -]ol
Kyoto University.
~--
(2)
(3) * On l e a v e of a b s e n c e from Department of P h y s i c s ,
05
20 (c)
1.81
--"
Fig. 1. (a) ~ x / %
C
o%
,.'o
%
-~.o0
: 0.4, (b) ~ x / %
: 0.1, (c) ~ x / ~ o = 0.0.
Volume 26A, number 3
P HY S I C S L E T T E R S
The condition (2) m e a n s that the c u r r e n t through the bulk ring is equal to the J o s e p h s o n c u r r e n t a c r o s s the link [1]. The p a r a m e t e r ~ can be c o n t r o l e d e i t h e r by adjusting the n o r m a l r e s i s t a n c e of the link o r by v a r y i n g the t e m p e r a t u r e T. R e c e n t l y S i l v e r and Z i m m e r m a n [3] have e x p e r i m e n t a l l y studied the o f a N b r i n g ( L ~ 0 . 4 c m ) at T = 4°K. They have a d j u s t e d the r e s i s t a n c e of the contact so t h a t a ~ 1 (E o ~ 20°K). F o r a > 1, the solution of eq. (2) is unique and G has a single m i n i m u m . We then obtain a r e v e r s i b l e m a g n e t i z a t i o n in the s t a t i c l i m i t . F o r a < 1, G has a n u m b e r of m i n i m a and m a x i m a [1]. In fig. 1, we show the p o t e n t i a l s f o r a = 0.22; f i g u r e s a t t a c h e d to the points A, B, C i n d i c a t e t h e i r r e l a t i v e heights in units of E o. Suppose we s t a r t with the s t at e A in fig. l a . With d e c r e a s i n g ~ x , the s t ab i l i t y of this state d e c r e a s e s ; f o r ~I'x < ½ ~ o , the state A b e c o m e s m e t a s t a b l e and eventually c o i n c i d e s with the state B (fig. lc). F r o m the t h e r m o d y n a m i c a l point of view, one e x p e c t s the i r r e v e r s i b l e t r a n s i t i o n A ~ C to o c c ur on d e c r e a s i n g @x f u r t h e r . Both r e v e r s i b l e and i r r e v e r s i b l e m a g n e t i z a tion c u r v e s have been o b s e r v e d by S i l v e r and Z i m m e r m a n , and t h e i r data can be a c c o u n te d for at l e a s t q u a l i t a t i v e l y by our t h e r m o d y n a m i c a l t h e o r y , which is s o m e w h a t s i m p l e r than the mode l p r o p o s e d by t h e m .
1 January 1968
So far, we have i g n o r e d fluctuations. When a v o l t ag e e x i s t s a c r o s s the link, t h e r e flows a d i s s i p a t i v e c u r r e n t , and c o n v e r s e l y , on the ground of Nyquist t h e o r e m , we expect a f l u c t u a tion of the v o l t a g e , which will d r i v e the f l u c t u a tion of the phase through the well-known J o s e p h son equation [2]. The fluctuation b e c o m e s p a r t i c u l a r l y significant, when the potential : b a r r i e r AG between two m i n i m a is c o m p a r a b l e with kT. Such a situation i s shown by fig. Ib, w h er e AG = 0.1 E o. F o r the weak link of ref. 4, ot = 0.22 would c o r r e s p o n d to E o ~ 90°K, and the c h a r a c t e r i s t i c f r e q u e n c y ¢o of the state A in fig. l b would be of o r d e r of 109sec -1 (this depends on the capacity). Hence the t r a n s i t i o n r a t e w exp {-AG/kT} might be c o m p a r a b l e with the modulation f r e q u e n c y (30Mc/s) of @x u s e d in the e x p e r i m e n t . Anyway, b e f o r e r e a c h i n g the c r i t i cal point shown by fig. l c , the p e r s i s t e n t c u r r e n t will s t a r t to decay through fluctuation, and the decay must~be d e t e c t e d e x p e r i m e n t a l l y by v a r y ing the f r e q u e n c y .
References 1. A. M. Goldman, P . J . Kreisman and D. J. Sealapino, Phys. Rev. Letters 13 (1965) 495. 2. B.D. Josephson, Phys. Letters 1 (1962) 251; Rev. Mod. Phys. 36 (1964) 216. 3. A.H. Silver and J. E. Zimmerman, Phys. Rev. 157 (1967) 317.
107
PHYSICS
1. I . M . K h a l a t n i k o v . Z h . E k s p . i T e o r F i z . 22 (1952) 687. 2. L . J . C h a l l i s , P r o c . P h y s . Soc. 80 (1962) 759. 3. K u a n g W e y - Y e n . Zh. E k s p . i T e o r . F i z . 42 (1962) 921; Soviet P h y s . J E T P 15 (1962) 635. 4. R . C . J o h n s o n a n d W . A . L i t t l e , P h y s , R e v . 130 {1963) 596. 5. D . A . N e e p e r a n d J . R . D i l l i n g e r , P h y s . R e v . 135 (1964) A1028,
ON
THE
LETTERS
1 J a n u a r y 1968
6. L . J . C h a l l i s a n d J . D , N . Cheeke. P r o g r e s s in r e f r i geration science and technology {Pergamon P r e s s . for the International Institute of Refrigeration) 1 (1965) 227, 7, D . A . N e e p e r , D . C . P e a r c e a n d R . M . W a s i l i k , Phys. Rev. 156 (1967) 764. 8. J . I . G i t t l e m a n and S.Bozu~,ski. Phys. Rev. 128 {1962) 646, 9. R . C . J o h n s o n , Bull. Am, Phys. Soc. 9 {1964) 713.
THERMODYNAMICS OF SUPERCONDUCTING
"WEAKLY RINGS
CONNECTED
S. N A K A J I M A a n d Y. K U R O D A * Institute for Solid State Physics, University of Tokyo, Minato-ku, Tokyo
Received 1 D e c e m b e r 1967
A t h e r m o d y n a m i c a l analysis is made of the magnetization of weakly connected superconducting rings as o b s e r v e d by Silver and Z i m m e r m a n . A possible decay of the p e r s i s t e n t c u r r e n t is predicted.
W h e n a w e a k l i n k (point c o n t a c t o r t u n n e l j u n c tion) is introduced into a superconducting ring [1], t h e j u m p in t h e p h a s e of t h e o r d e r p a r a m eter across the link is no longer an integral mult i p l e of 2~. If t h e r i n g i s t h i c k c o m p a r e d w i t h t h e penetration depth and the link is so small that the magnetic flux through it may be ignored, the p h a s e j u m p i s g i v e n by ( 2 1 r ~ / ~ o ) , w h e r e 4~ i s t h e f l u x e n c l o s e d b y t h e r i n g a n d ~ o = (~/~c/e). In t h e p r e s e n c e of t h e e x t e r n a l f l u x 4)x, t h e G i b b s p o t e n t i a l of o u r s y s t e m t a k e s t h e f o r m
3 0-9~. 20
(a)
10 00 3 50 05
20 A
G : GO + (1/2L)(~-4)x)2
- E o c o s ( 2 ~ / 4 ) o)
(1)
(b)
Here Go is a 4)-independent part, L is the induct a n c e of t h e b u l k r i n g , a n d t h e l a s t t e r m i s t h e p h a s e c o u p l i n g e n e r g y w h i c h r e s u l t s in t h e J o s e p h s o n c u r r e n t [2] a c r o s s t h e l i n k . In t h e s t a t i c l i m i t , tim s y s t e m m u s t b e i n t h e t h e r m o dynamic state for which G is a minimum with r e s p e c t to ~): sin[2~/4~o]
= - 27rot[((b/~o) - ( 4 ) x / 4 ) o ) ]
10
oo
106
-0.96
Io
oo ~" o'~/ -]ol
Kyoto University.
~--
(2)
(3) * On l e a v e of a b s e n c e from Department of P h y s i c s ,
05
20 (c)
1.81
--"
Fig. 1. (a) ~ x / %
C
o%
,.'o
%
-~.o0
: 0.4, (b) ~ x / %
: 0.1, (c) ~ x / ~ o = 0.0.
Volume 26A, number 3
P HY S I C S L E T T E R S
The condition (2) m e a n s that the c u r r e n t through the bulk ring is equal to the J o s e p h s o n c u r r e n t a c r o s s the link [1]. The p a r a m e t e r ~ can be c o n t r o l e d e i t h e r by adjusting the n o r m a l r e s i s t a n c e of the link o r by v a r y i n g the t e m p e r a t u r e T. R e c e n t l y S i l v e r and Z i m m e r m a n [3] have e x p e r i m e n t a l l y studied the o f a N b r i n g ( L ~ 0 . 4 c m ) at T = 4°K. They have a d j u s t e d the r e s i s t a n c e of the contact so t h a t a ~ 1 (E o ~ 20°K). F o r a > 1, the solution of eq. (2) is unique and G has a single m i n i m u m . We then obtain a r e v e r s i b l e m a g n e t i z a t i o n in the s t a t i c l i m i t . F o r a < 1, G has a n u m b e r of m i n i m a and m a x i m a [1]. In fig. 1, we show the p o t e n t i a l s f o r a = 0.22; f i g u r e s a t t a c h e d to the points A, B, C i n d i c a t e t h e i r r e l a t i v e heights in units of E o. Suppose we s t a r t with the s t at e A in fig. l a . With d e c r e a s i n g ~ x , the s t ab i l i t y of this state d e c r e a s e s ; f o r ~I'x < ½ ~ o , the state A b e c o m e s m e t a s t a b l e and eventually c o i n c i d e s with the state B (fig. lc). F r o m the t h e r m o d y n a m i c a l point of view, one e x p e c t s the i r r e v e r s i b l e t r a n s i t i o n A ~ C to o c c ur on d e c r e a s i n g @x f u r t h e r . Both r e v e r s i b l e and i r r e v e r s i b l e m a g n e t i z a tion c u r v e s have been o b s e r v e d by S i l v e r and Z i m m e r m a n , and t h e i r data can be a c c o u n te d for at l e a s t q u a l i t a t i v e l y by our t h e r m o d y n a m i c a l t h e o r y , which is s o m e w h a t s i m p l e r than the mode l p r o p o s e d by t h e m .
1 January 1968
So far, we have i g n o r e d fluctuations. When a v o l t ag e e x i s t s a c r o s s the link, t h e r e flows a d i s s i p a t i v e c u r r e n t , and c o n v e r s e l y , on the ground of Nyquist t h e o r e m , we expect a f l u c t u a tion of the v o l t a g e , which will d r i v e the f l u c t u a tion of the phase through the well-known J o s e p h son equation [2]. The fluctuation b e c o m e s p a r t i c u l a r l y significant, when the potential : b a r r i e r AG between two m i n i m a is c o m p a r a b l e with kT. Such a situation i s shown by fig. Ib, w h er e AG = 0.1 E o. F o r the weak link of ref. 4, ot = 0.22 would c o r r e s p o n d to E o ~ 90°K, and the c h a r a c t e r i s t i c f r e q u e n c y ¢o of the state A in fig. l b would be of o r d e r of 109sec -1 (this depends on the capacity). Hence the t r a n s i t i o n r a t e w exp {-AG/kT} might be c o m p a r a b l e with the modulation f r e q u e n c y (30Mc/s) of @x u s e d in the e x p e r i m e n t . Anyway, b e f o r e r e a c h i n g the c r i t i cal point shown by fig. l c , the p e r s i s t e n t c u r r e n t will s t a r t to decay through fluctuation, and the decay must~be d e t e c t e d e x p e r i m e n t a l l y by v a r y ing the f r e q u e n c y .
References 1. A. M. Goldman, P . J . Kreisman and D. J. Sealapino, Phys. Rev. Letters 13 (1965) 495. 2. B.D. Josephson, Phys. Letters 1 (1962) 251; Rev. Mod. Phys. 36 (1964) 216. 3. A.H. Silver and J. E. Zimmerman, Phys. Rev. 157 (1967) 317.
107