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Submitted on 1 Jan 1978
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CHARACTERISTIC RESULTING FROM THE
WERTHAMER EQUATION AT FINITE
TEMPERATURES AND CAPACITANCE
W. Schlup
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, supplkmenr au no 8, Tome 39, aoirt 1978, page
C6-565
C H A R A C T E R I S T I C R E S U L T I N G F R O M T H E W E R T H A M E R E Q U A T I O N A T F I N I T E T E M P E R A T U R E S A N D
C A P A C I T A N C E
W.A. Schlup
IBM
Zurich Research Laboratory,
8803REschZikon
ZH,
SwitzerZand
Rbsumb.- L'bquation de Werthamer e s t r b s o l u e pour une j o n c t i o n tunnel synunbtrique e t d e p e t i t e dimension, sous c o u r a n t c o n s t a n t , pour d i f f b r e n t e s c a p a c i t d s e t 1 tempbrature f i n i e . Les c a r a c t b r i s t i q u e s moyennes tension-courant s o n t d i s c u t b e s . On trouve une s t r u c t u r e d e frdquences subharmoniques d ' o r d r e impair formant une s b r i e d e p o i n t s d e rebroussement. A b s t r a c t . - The Werthamer e q u a t i o n f o r a d c c u r r e n t - b i a s e d small t u n n e l j u n c t i o n between
i d e n t i c a l superconductors i s solved a t f i n i t e t e m p e r a t u r e and f o r v a r i o u s c a p a c i t a n c e s . The average v o l t a g e - c u r r e n t c h a r a c t e r i s t i c s a r e i n v e s t i g a t e d ; c u s p l i k e s t r u c t u r e s a r e found a t odd subharmonic gap v o l t a g e s .
The d c I-V c h a r a c t e r i s t i c s of a c u r r e n t - biased c a p a c i t i v e Jpsephson j u n c t i o n h a s been in- v e s t i g a t e d b o t h e x p e r i m e n t a l l y / 1 , 2 / and theore- t i c a l l y / 3 , 4 / . As a t h e o r e t i c a l model, t h e Werthaner e q u a t i o n has been used, d e r i v e d m i c r o s c o p i c a l l y from t h e weak coupling BCS t h e o r y / 5 / . I t i s an i n t e g r o - d i f f e r e n t i a l e q u a t i o n f o r t h e time evolu- t i o n of t h e phase ( d i f f e r e n c e ) which t a k e s t h e dynamics of t h e v o l t a g e f u l l y i n t o account i n con-
t r a s t t o a l l d i f f e r e n t i a l - t y p e Josephson e q u a t i o n s which make u s e of t h e a d i a b a t i c approximation. The most s t r i k i n g d i s c r e p a n c y between a t a l l frequen-
c i e s w = w / n , n = 1 , 2, 3 ,
...
(U gap f r e q u e n c y ) ,n g g
whereas Werthamer's e q u a t i o n p r e d i c t s i t f o r odd v a l u e s n = 1 , 3, 5 ,
...
o n l y161.
The purpose of t h i s c o n t r i b u t i o n i s t o extend t h e s o l u t i o n of t h e Werthamer e q u a t i o n , which s o f a r has been solved f o r temperature
T
= 0and v a r i o u s c a p a c i t a n c e s 131, o r f o r c a p a c i t a n c e C = 0 a t t h e reduced temperatures T/Tc = 0.5, 0 . 8 , 0.95 / 4 / . We p r e s e n t r e s u l t s f o r temperatures cor- responding t o A(T)/T = 0 , 0 . 5 , 1 , 1.5 o r TITc =
1 , 0.974, 0.905, 0.813, and v a r i o u s v a l u e s of t h e c a p a c i t a n c e parameter
6 = 2A(T)C/6GN [A(T) gap energy, GN normal conduc- t a n c d
.
Werthamer's e q u a t i o n f o r a small j u n c t i o n The F o u r i e r transforms of G ( t ) o r F ( t ) d e f i n e t h e t u n n e l f u n c t i o n s Io(w),
J0(w) o r I I (w),
r e s p e c t i - v e l y . For I < J , ( O ) , t h e s t a b l e ( s t a t i c ) s o l u t i o n i sOS
= a r c s i n I / J l ( 0 ) ; f o r I > J (0) i t t r a n s f o r m 1 i n t o a p e r i o d i c s t e a d y - s t a t e s o l u t i o n w i t h $ ( t + h / 2eV) = $ ( t )+
2n. T h i s s o l u t i o n c o e x i s t s w i t h t h e s t a t i c s o l u t i o n i n t h e range between a b i f u r c a t i o n c u r r e n t I and J 1 ( 0 ) , t h e maximum Josephson cur- r e n t . The numerical s o l u t i o n i s found by a F o u r i e r expansion method of O ( t ) , s t a r t i n g from a g i v e n (dc average v o l t a g e ) V and F o u r i e r c o e f f i c i e n t s {Cn}, which a r e improved by i t e r a t i o n ; t h e impro-I
ved s o l u t i o n {Cn) i s determined by s o l v i n g t h e d i f f e r e n t i a l e q u a t i o n ( I ) , where t h e i n t e g r a l eva- l u a t e d by means of t h e s t a r t i n g f u n c t i o n $ ( t ) i s considered a s an inhomogeneity. The i t e r a t i o n i s continued u n t i l r e l e v a n t d i g i t s i n I and iCN} no longer change, even i f t h e ( f i n i t e ) number of F o u r i e r c o e f f i c i e n t s i s i n c r e a s e d f u r t h e r .
The s t e a d y - s t a t e c u r r e n t I can be s e p a r a t e d i n t o a q u a s i p a r t i c l e p a r t I generated by t h e
4
Q
term, t h e l i n e a r p a r t of Io(w), and G+, and i n t o t h e p a i r currerit Ip stemming from F i t ) . For a pe- r i o d i c f u n c t i o n
4
(t),
t h e displacement c u r r e n t v a n i s h e s on t h e average and t h e r e f o r e t h e t o t a l ( i n p u t ) c u r r e n t s e p a r a t e s i n t o i s I + I = IQ
P (2)cM
G6
-
$ +
%
( + d t l [ ~ + ( t ' ) s i n b-'(i-t') + F ( t ) F i g u r e 1 shows t h e c h a r a c t e r i s t i c s I (V),
2eQ
I (V)
,
I(V) f o r TITc = 0.813 and t h e c a p a c i t a n c e Pparameter 6 = 0.1. The d o t t e d curve r e p r e s e n t s where t h e k e r n e l s F ( t ) and G+(t) = G ( t ) + (GNH/e)
10(w = eV/h), which i s i d e n t i c a l w i t h I and I 6 ' ( t ) a s y m p t o t i c a l l y o s c i l l a t e w i t h t h e gap f r e - f o r 8 =
-.
For V-
10, I I 'L G V, whereas t h eQ
quency w and decay l i k e t-l ( s e e r e f e r e n c e
1 7 1 .
Q
Ng
p a i r c u r r e n t v a n i s h e s i n agreement w i t h t h e asymp- t o t i c s o l u t i o n of (1) 181. For eV/A
2
2 , I and IP have a s i n g u l a r i t y , whereas f o r e V / A % 213, 215, t h e c u r v e s e x h i b i t f i n i t e c u s p s w i t h an I minimum; t h e q u a s i p a r t i c l e c u r r e n t I behaves analogous t o
Q
I (V) b u t shows a s t e p l i k e s t r u c t u r e a t odd subhar- 0
monic f r e q u e n c i e s . On t h e whole, I(V)
,
which b e a u s e of t h e d i s c o n t i n u i t y i n Io(V) d r o p s r a p i d l y j u s t below eV/A = 2 , v a r i e s o n l y s l i g h t l y a t lower v o l - t a g e s , b u t must d e c r e a s e a g a i n a s V + 0 , s i n c e t h e v o l t a g e should r e s e t a t t h e b i f u r c a t i o n c u r r e n t I(6) < J 1 (0) = 0.998 %Ale.
c l e a r l y e x h i b i t t h e R i e d e l s i n g u l a r i t y a t t h e gap v o l t a g e and c u s p l i k e c u r r e n t minima a t odd subhar- monic gap f r e q u e n c i e s ; when t e m p e r a t u r e s a r e s m a l l , t h e subharmonic gap s t r u c t u r e becomes more s t e p l i h .
F i g . 2 : The c h a r a c t e r i s t i c s I(V) and Ip(V) ( i n - dexed P) f o r
6
= 1 a t v a r i o u s t e m p e r a t u r e s w i t hA ( T ) / T = o(--), 0.5 (-.-),
(-.-I,
I (-.-),1.5 (-
...-
), and (....); o r i g i n s a r e d i s p l a c e d and maximum Josephson c u r r e n t s a r e i n d i c a t e d by c i r c l e s .F i g . 1 : The c h a r a c t e r i s t i c s of t h e d c c u r r e n t - b i a - sed s m a l l t u n n e l j u n c t i o n a t TITc = 0.813 f o r ca- p a c i t a n c e parameter 6 = ZA(T)C/HG = 0.1 s e p a r a t e d i n t o q u a s i p a r t i c l e I and p a i r I * c o n t r i b u t i o n .
Q
P I n c o n t r a s t , I f o l l o w s 10(V) c l o s e l y b u t , i n ge- .Q
n e r a l , d o e s n o t v a n i s h f o r V-rO.I n f i g u r e 2, I(V) and I (V) a r e compared P
f o r 6' 1 a t d i f f e r e n t temperatures, namely, f o r A(T) /T e q u a l t o 0 (dashed c u r v e ) , 0.5 ( d a s h , one- d o t c u r v e ) , I ( d a s h , two-dot c u r v e ) , 1,5 ( d a s h , t h r e e - d o t c u r v e ) and ( d o t t e d c u r v e ) . The cor- r e s p o n d i n g o r i g i n s a r e d i s p l a c e d a l o n g t h e c u r r e n t a x i s . The r e s p e c t i v e maximum Josephson c u r r e n t s a r e i n d i c a t e d by f u l l c i r c l e s [ i n c r e a s i n g f o r i n c r e a - s i n g A ( T ) / ~ . The c u r r e n t e x h i b i t s a s i n g u l a r i t y a t t h e g a p v o l t a g e and cusps a t odd subharmonic gap f r e q u e n c i e s , becoming s t e p l i k e when T = 0 i s approa- ched. The p a i r c u r r e n t (P) i s n e g a t i v e f o r l a r g e V i n agreement w i t h t h e a s y m p t o t i c s o l u t i o n 181.
I n c o n c l u s i o n , i t can b e s a i d t h a t d c I-V c h a r a c t e r i s t i c s computed from t h e Werthamer equation
Apart from t h e s e d e t a i l s , t h e c h a r a c t e r i s t i c s f o l - low 10(V) f o r V > V b u t d e c r e a s e almost l i n e a r l y g' f o r t e m p e r a t u r e s n e a r T f o r 0 < V < V o r remain g ' p r a c t i c a l l y c o n s t a n t i n t h a t r a n g e a t low tempera- t u r e s . R e f e r e n c e s
/ I / Yanson, I . K . , S v i s t u n o v , V.M., and Dmitrenka,
I . M . , Sov.Phys. JETP (1965) 1404.
/ 2 / G i a e v e r , I., and Z e l l e r , H.R., Phys. Rev. B
1
(1970) 4278./3/ McDonald, D.G., Johnson, E.G.and H a r r i s , R.E.,
Phys. Rev. B
12
(1976) 1028./ 4 / Z o r i n , A . B . , and L i k h a r e v , K.K. SOY. J. Low
Temp. Phys.
3
(1977) 70./ 5 / Werthamer, N.R., Phys. Rev.
147
(1966) 255./ 6 / H a s s e l b e r g , L.E., L e v i n s e n , M.T.and Samuelsen, M.R., J. Low Temp. Phys.
