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Resistive transition tailing in an array of Josephson junctions
Volume 28A, number 1
P HYSIC S L E T T E RS
T h e t h e o r e t i c a l b i n d i n g e n e r g y , w h i c h i s the s u m of t h e t h r e e t e r m s c o m p u t e d a b o v e , i s - 4 . 1 2 Ry p e r ion w h i c h i s in v e r y g o o d a g r e e m e n t with the experimental value. Thus, the modificat i o n s i n t r o d u c e d h e r e i m p r o v e t h e v a l u e of t h e b i n d i n g e n e r g y of A1 c o n s i d e r a b l y . W h e t h e r t h e s e m o d i f i c a t i o n s w i l l g i v e t h e e n e r g y m i n i m u m at o r n e a r t h e o b s e r v e d l a t t i c e s p a c i n g i s not y e t known.
21 October 1968
1. W.A.Harrison, Phys. Rev. 136 (1964) Al107; Pseudopotentials in the theory of metals (W, A. Benjamin, Inc. New York. 1966). 2. H.Brooks, Trans. AIME 277 (1963) 546. 3. W. Kohn and L . J . S h a m , Phys. Rev. 140 (1965) Al133. 4. A.O, E.Animalu, Proc. Roy. Soc. (London) A294 (1966) 376. 5. J . C . S l a t e r , Phys. Rev. 81 (1951) 385. 6. M.Gell-Mann and K.A.Brueckner, Phys. Rev. 106 (1957) 364. 7. V. Heine and I.Abarenkov. Phil. Mag. 9 (1964) 451. 8. A . O . E . Animalu, F. Bonsignori and V. Borotolani, Nuovo Cimento 44 (1966) 159.
* * * * *
RESISTIVE
TRANSITION
TAILING
IN
AN
ARRAY
OF
JOSEPHSON
JUNCTIONS
T. D. C L A R K * Mullard Research Laboratories, Redhill, Surrey, UK
and D. R. T I L L E Y Physics Department, University of Essex, Colchester, Essex, UK Received 6 September 1968
The resistive transitions of arrays of tin spheres have a low temperature tail determined by phase slippage at the contact points.
O n e of u s h a s r e c e n t l y d e s c r i b e d [1] e x p e r i m e n t s in w h i c h a r r a y s of tin s p h e r e s p r e s s e d t o g e t h e r c a r r i e d s u p e r c u r r e n t s . T h e v a r i a t i o n of s u p e r c u r r e n t with t e m p e r a t u r e , and t h e s t r u c t u r e of t h e d i f f e r e n t i a l c o n d u c t a n c e - v o l t a g e c h a r a c t e r i s t i c s , i n d i c a t e d J o s e p h s o n t u n n e l l i n g [2] a c r o s s t h e p o i n t s of c o n t a c t b e t w e e n t h e s p h e r e s . F i g . l a s h o w s r e s i s t i v e t r a n s i t i o n s of an a r r a y of 1 m m tin b a l l s , p r e s s e d in a t u b e 9 c m l o n g and ½ c m in d i a m e t e r . T h e r e s i s t a n c e of t h e a r r a y w a s 40 ~2 at 4 . 2 ° K , and t h e z e r o v o l t a g e s u p e r c u r r e n t w a s 470 ~ A at 1.3°K. It can be s e e n that the t r a n s i t i o n s a r e f a i r l y b r o a d , and t a i l s l o w l y into t h e v o l t a g e n o i s e l e v e l at t h e low t e m p e r a t u r e end. W e w i s h to s h o w t h a t t h i s t a i l i n g can be u n d e r s t o o d in t e r m s of t h e r m a l l y a c t i v a t e d s l i p p a g e of t h e o r d e r p a r a m e t e r , as s u g g e s t e d by A n d e r son [3]. C o n s i d e r a r e s i s t a n c e t r a n s i t i o n m e a s u r e d on a s i n g l e J o s e p h s o n j u n c t i o n at f i x e d c u r r e n t Je" * Present address: Philips Research Laboratories, Eindhoven, The Netherlands.
F o l l o w i n g A n d e r s o n [3], w e w r i t e the e n e r g y of the junction as E =Jo c o s ~ - Je ~
where ~ is the phase difference across the junction. For Je < Jo the phase will change by thermal activation to give a voltage V d.c. cc dda/dt oc exp (- A E / k T )
(2)
with a c t i v a t i o n e n e r g y A E = 2 J o (1 - j 2 / j ~ o ) ½ _ lrJe + 2 j e a r c s i n ( J e / J o )" (3) F o r J e << J o , A E : 2 J o - ~rge .
(4)
For smaller Jo, the activation energy is smaller than eq. (4). C o m b i n i n g e q s . (2) and (4) with t h e e x p e c t e d v a r i a t i o n of J o with t e m p e r a t u r e , J o = = J o o (1 - T / T c ) n e a r T c [4], we s e e t h a t t h e r e s i s t a n c e at f i x e d m e a s u r i n g c u r r e n t s h o u l d s a t i s fy
In (R / R N) : 2Joo / k T c - (2Joo - ~ZJe)/kT . (5) n e a r to T c.
62
(1)
Volume28A, number 1
1.0
PHYSICS LETTERS
R/R n
21 October 1968
1.0
%. Measuring currents •
X
MeaSuring currentS: X 4HA ~)~3 + 101JA zx 18pA
x rn + A.
4~ spa lo/J 18~A
0-I. 0"4
/
~5
./
'
3'-7o
i
l
l
T(OK ~
O0 1.00
I-(~
1102
1-53
"rr/T
Fig. i. Resistive transitions of a tube of tin balls, la. R/R N versus T; lb. Logarithmic plot of R/R N versus Tc/T.
Fig. Ib shows logarithmic plots of R/R N against T c / T for the three transitions of fig. la, and for one other transition. T h e arrows on both figures m a r k the point at which the Josephson current, extrapolated from the low temperature value [4], is equal to the measuring current; the phase slippage rlegions are those with temperature below the arrow. It can be seen from fig. lb that the g e n e r a l a g r e e m e n t of our data with eqs. (2) and (5) i s good. The slopes of the g r a p h s i n c r e a s e somewhat with d e c r e a s i n g t e m p e r a t u r e , as a c o m p a r i s o n of eqs. (3) and (4) i m p l i e s . As the applied c u r r e n t J e i n c r e a s e s , the slope of the s t r a i g h t l i n e p a r t d e c r e a s e s , as p r e d i c t e d by eq. (5); the d i f f e r e n c e s between the slopes a g r e e with the d i f f e r e n c e s between the m e a s u r i n g c u r r e n t s , followin~ eq. (5). T h e o r e t i c a l 15] and e x p e r i m e n t a l [6,7] work has been done on g r a n u l a r s u p e r c o n d u c t o r s , thin f i l m s in which it i s a s s u m e d that t h e r e a r e J o s e p h s o n j u n c t i o n s (weak links) between the
g r a i n s . In s o m e ways a r r a y s of s p h e r e s f o r m a m o d e l of a g r a n u l a r s u p e r c o n d u c t o r , and the f l u c t u a t i o n s below T c in our a r r a y a r e the s a m e a s those d i s c u s s e d qualitatively in the second half of the paper of Strongin et al. [7]. W e should like to thank Dr. A. K. Niessen for a helpful discussion.
References 1. T.D. Clark, Phys. Letters 27A (1968) 585. 2. B.D.Josephson. Phys. Letters 1 (1962) 251. 3. P.W.Anderson, Prog. Low Temp. Phys. 5 (1967) i. §3. 4. V.Ambegaokar and A. Baratoff, Phys. Rev. Letters 10 (1963) 486; 11 (1963) 104. 5. R.H. Parmenter, Phys. Rev. 154 (1967) 353. 6. R.E.Glover. Phys. Letters 25A (1967) 42. 7. M. Strongin, O. F. Kammerer. J. Crow, R.S. Thompson and H. L. Fine, Phys. Rev. Letters 20 (1968) 922.
63
P HYSIC S L E T T E RS
T h e t h e o r e t i c a l b i n d i n g e n e r g y , w h i c h i s the s u m of t h e t h r e e t e r m s c o m p u t e d a b o v e , i s - 4 . 1 2 Ry p e r ion w h i c h i s in v e r y g o o d a g r e e m e n t with the experimental value. Thus, the modificat i o n s i n t r o d u c e d h e r e i m p r o v e t h e v a l u e of t h e b i n d i n g e n e r g y of A1 c o n s i d e r a b l y . W h e t h e r t h e s e m o d i f i c a t i o n s w i l l g i v e t h e e n e r g y m i n i m u m at o r n e a r t h e o b s e r v e d l a t t i c e s p a c i n g i s not y e t known.
21 October 1968
1. W.A.Harrison, Phys. Rev. 136 (1964) Al107; Pseudopotentials in the theory of metals (W, A. Benjamin, Inc. New York. 1966). 2. H.Brooks, Trans. AIME 277 (1963) 546. 3. W. Kohn and L . J . S h a m , Phys. Rev. 140 (1965) Al133. 4. A.O, E.Animalu, Proc. Roy. Soc. (London) A294 (1966) 376. 5. J . C . S l a t e r , Phys. Rev. 81 (1951) 385. 6. M.Gell-Mann and K.A.Brueckner, Phys. Rev. 106 (1957) 364. 7. V. Heine and I.Abarenkov. Phil. Mag. 9 (1964) 451. 8. A . O . E . Animalu, F. Bonsignori and V. Borotolani, Nuovo Cimento 44 (1966) 159.
* * * * *
RESISTIVE
TRANSITION
TAILING
IN
AN
ARRAY
OF
JOSEPHSON
JUNCTIONS
T. D. C L A R K * Mullard Research Laboratories, Redhill, Surrey, UK
and D. R. T I L L E Y Physics Department, University of Essex, Colchester, Essex, UK Received 6 September 1968
The resistive transitions of arrays of tin spheres have a low temperature tail determined by phase slippage at the contact points.
O n e of u s h a s r e c e n t l y d e s c r i b e d [1] e x p e r i m e n t s in w h i c h a r r a y s of tin s p h e r e s p r e s s e d t o g e t h e r c a r r i e d s u p e r c u r r e n t s . T h e v a r i a t i o n of s u p e r c u r r e n t with t e m p e r a t u r e , and t h e s t r u c t u r e of t h e d i f f e r e n t i a l c o n d u c t a n c e - v o l t a g e c h a r a c t e r i s t i c s , i n d i c a t e d J o s e p h s o n t u n n e l l i n g [2] a c r o s s t h e p o i n t s of c o n t a c t b e t w e e n t h e s p h e r e s . F i g . l a s h o w s r e s i s t i v e t r a n s i t i o n s of an a r r a y of 1 m m tin b a l l s , p r e s s e d in a t u b e 9 c m l o n g and ½ c m in d i a m e t e r . T h e r e s i s t a n c e of t h e a r r a y w a s 40 ~2 at 4 . 2 ° K , and t h e z e r o v o l t a g e s u p e r c u r r e n t w a s 470 ~ A at 1.3°K. It can be s e e n that the t r a n s i t i o n s a r e f a i r l y b r o a d , and t a i l s l o w l y into t h e v o l t a g e n o i s e l e v e l at t h e low t e m p e r a t u r e end. W e w i s h to s h o w t h a t t h i s t a i l i n g can be u n d e r s t o o d in t e r m s of t h e r m a l l y a c t i v a t e d s l i p p a g e of t h e o r d e r p a r a m e t e r , as s u g g e s t e d by A n d e r son [3]. C o n s i d e r a r e s i s t a n c e t r a n s i t i o n m e a s u r e d on a s i n g l e J o s e p h s o n j u n c t i o n at f i x e d c u r r e n t Je" * Present address: Philips Research Laboratories, Eindhoven, The Netherlands.
F o l l o w i n g A n d e r s o n [3], w e w r i t e the e n e r g y of the junction as E =Jo c o s ~ - Je ~
where ~ is the phase difference across the junction. For Je < Jo the phase will change by thermal activation to give a voltage V d.c. cc dda/dt oc exp (- A E / k T )
(2)
with a c t i v a t i o n e n e r g y A E = 2 J o (1 - j 2 / j ~ o ) ½ _ lrJe + 2 j e a r c s i n ( J e / J o )" (3) F o r J e << J o , A E : 2 J o - ~rge .
(4)
For smaller Jo, the activation energy is smaller than eq. (4). C o m b i n i n g e q s . (2) and (4) with t h e e x p e c t e d v a r i a t i o n of J o with t e m p e r a t u r e , J o = = J o o (1 - T / T c ) n e a r T c [4], we s e e t h a t t h e r e s i s t a n c e at f i x e d m e a s u r i n g c u r r e n t s h o u l d s a t i s fy
In (R / R N) : 2Joo / k T c - (2Joo - ~ZJe)/kT . (5) n e a r to T c.
62
(1)
Volume28A, number 1
1.0
PHYSICS LETTERS
R/R n
21 October 1968
1.0
%. Measuring currents •
X
MeaSuring currentS: X 4HA ~)~3 + 101JA zx 18pA
x rn + A.
4~ spa lo/J 18~A
0-I. 0"4
/
~5
./
'
3'-7o
i
l
l
T(OK ~
O0 1.00
I-(~
1102
1-53
"rr/T
Fig. i. Resistive transitions of a tube of tin balls, la. R/R N versus T; lb. Logarithmic plot of R/R N versus Tc/T.
Fig. Ib shows logarithmic plots of R/R N against T c / T for the three transitions of fig. la, and for one other transition. T h e arrows on both figures m a r k the point at which the Josephson current, extrapolated from the low temperature value [4], is equal to the measuring current; the phase slippage rlegions are those with temperature below the arrow. It can be seen from fig. lb that the g e n e r a l a g r e e m e n t of our data with eqs. (2) and (5) i s good. The slopes of the g r a p h s i n c r e a s e somewhat with d e c r e a s i n g t e m p e r a t u r e , as a c o m p a r i s o n of eqs. (3) and (4) i m p l i e s . As the applied c u r r e n t J e i n c r e a s e s , the slope of the s t r a i g h t l i n e p a r t d e c r e a s e s , as p r e d i c t e d by eq. (5); the d i f f e r e n c e s between the slopes a g r e e with the d i f f e r e n c e s between the m e a s u r i n g c u r r e n t s , followin~ eq. (5). T h e o r e t i c a l 15] and e x p e r i m e n t a l [6,7] work has been done on g r a n u l a r s u p e r c o n d u c t o r s , thin f i l m s in which it i s a s s u m e d that t h e r e a r e J o s e p h s o n j u n c t i o n s (weak links) between the
g r a i n s . In s o m e ways a r r a y s of s p h e r e s f o r m a m o d e l of a g r a n u l a r s u p e r c o n d u c t o r , and the f l u c t u a t i o n s below T c in our a r r a y a r e the s a m e a s those d i s c u s s e d qualitatively in the second half of the paper of Strongin et al. [7]. W e should like to thank Dr. A. K. Niessen for a helpful discussion.
References 1. T.D. Clark, Phys. Letters 27A (1968) 585. 2. B.D.Josephson. Phys. Letters 1 (1962) 251. 3. P.W.Anderson, Prog. Low Temp. Phys. 5 (1967) i. §3. 4. V.Ambegaokar and A. Baratoff, Phys. Rev. Letters 10 (1963) 486; 11 (1963) 104. 5. R.H. Parmenter, Phys. Rev. 154 (1967) 353. 6. R.E.Glover. Phys. Letters 25A (1967) 42. 7. M. Strongin, O. F. Kammerer. J. Crow, R.S. Thompson and H. L. Fine, Phys. Rev. Letters 20 (1968) 922.
63