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Submitted on 1 Jan 1978
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NON EQUILIBRIUM BEHAVIOUR OF
SUPERCONDUCTING WEAK LINKS AT LOW
VOLTAGES
Yi-Han Kao
To cite this version:
Yi-Han Kao.
NON EQUILIBRIUM BEHAVIOUR OF SUPERCONDUCTING WEAK LINKS
JOURNAL DE PHYSIQUE Colloque C6, supplPmenr au no 8, Tome 39, aoiit 1978, page C6-552
Yi-Han Kao
Physics Department, S t a t e U n i v e r s i t y o f New York, Stony Brook, N.Y 11794, U.S.A.
R6sumQ.- On propose un modiile tenant compte de quelques propriQtQs hors Qquilibre de paires supraconductrices dans des ponts de petite dimension.
Abstract.
-
A model is proposed to account for some non-equilibrium properties of superconducting pairs in a weak link.1. INTRODUCTION.- The time-dependence of supercon- ducting pairs in a weak link has usually been in- terpreted on the basis of a simple resistively- shunted-junction (RSJ) model. More recently, a model
derived from the time-dependent Ginzburg-Landau (TDGL) equation has been suggested / l / in order to account for non equilibrium behaviour of the pairs. In the RSJT model /2/, / 3 / Jensen and Lindelof solved the one-dimensional'TDGL equation by analo- gous computation in which the (R/<)2 size-dependence of the superconducting weak link has been neglected. We vould like to show that, by including this size- dependence in the RSJr model, analytical solutions can actually be obtained. These solutions can be
conveniently employed to describe some dynamical properties of the superconducting pairs. It can also be shown that a "cos $" term should be present in both the supercurrent and the pair density.
2. MODEL.- We assume that the depairing effects are so strong that the order parameter J, in the weak link has become much smaller than its equilibrium value
eo
; for a very narrow weak link, the order parameter satisfies the following TDGL equation.where J, = $/qo, 5 is the GL coherence length, T is the pair relaxation time and U is the electroche- mica1 potential. If a voltage V has developed across
the weak link region 0 ( X
2
R, we assume thatp =
-
eVx/l. Solutions of equation ( 1 ) can be written as $(y,t) = g(y) + f (y) exp (iwot), where%
y = xja, wo = 2 eV/H and g(y) = f (I-y) satisfies
the following equation :
with 0 = w o ~ a 2 and a = R/<. The last term in (2) has been neglected in the RSJT model.
Using the "rigid" boundary condition, g (0) = 1, g(l) = 0, the low-voltage solutions of equation (2)
( W ~ T << 1)can be obtained by WKB approximation :
i0 y
--
i0sin (a(~-~) +
-
(1-~2))g(y) = e'4a2 4a i0 ( 3 )
sin (a
+
-) 4aThis analytical form can by used in the cal- culation of supercurrent and pair density at any point in the weak link.
3. DISCUSSION.- The supercurrent density Js (y,t) and pair density N (y,t) can be written in the
form :
js(y,t) = (y) sin $ + I1 (7) cos $ + I2 (Y) (ha) N~ (y,t) = N~ (y) sin $ + NI (y) cos
+
+ NP (Y) (4b)where $ = wot and Io, 11, 12, No, NI, N2 are func- tions of 0, a, y derived from (3).
A. I-V characteristics.- When the weak link (WL) is voltage-biased, the time-averaged supercurrent is given by I (y). At low voltages (wo << I), this
2
gives a linear I-V characteristic with conductance uV (y). The mean value of a temperature
-
and length- dependent conductance is :where i = (eH/mR)
l$o
(T)1
is the critical current for a very short weak link (a a: 1).If the WL con- tains a shunt resistance R and is current biased,'~esearch supported in part by the National Science Foundation.
t h e conductance then t a k e s t h e form o = 1 / R + oV.
I For a s h o r t WL, q approaches 1.2. No measurement of B. The c r i t i c a l c u r r e n t . - The c r i t i c a l c u r r e n t of
t h e WL i s g i v e n by I. = i o a / s i n a , ( s i n a # 0 ) . F o r a s h o r t WL, I.
-
i ( 1 + a 2 / 6 ) . i v a r i e s w i t h T a s (1-
TIT ) , t h e c o r r e c t i o n term g i v e s a n o t h e r f a c t o r of (1- TITc) a t temperature T
Tc.C. The supercurrent.- I1 (y) and I2 (y) a r e compli- c a t e d f u n c t i o n s of y. A f t e r averaged over t h e whole
WL, Js ( t ) c a n be w r i t t e n a s I. s i n $ + a V ( I + p c o s $ ) a t low v o l t a g e s , where t h e c o e f f i c i e n t of t h e "COS 4 t e r n " i s given by a 3 c o s a
-
3 s i n a + 3 a c o s a P = 2 a 3-
3 a + 3 s i n a c o s a (6) t h i s c o e f f i c i e n t i s known t o t h e a u t h o r a t t h e pre- s e n t time. The time v a r i a t i o n of N ( t ) i s d e p i c t e d i n f i g u r e 1.The dynamics of t h e p a i r s i s s e e n t o b e con- s i s t e n t w i t h a p h a s e s l i p p r o c e s s a t t h e c e n t r e of WL (y = 112). The s u p e r c u r r e n t r e v e r s e s i t s d i r e c - t i o n when t h e p a i r d e n s i t y v a n i s h e s momentarily. The l o c a l v e l o c i t y d i v e r g e s and changes s i g n correspon- d i n g t o a phase s l i p p a g e of 217. The whole p r o c e s s r e p e a t s i t s e l f a t t h e Josephson frequency.
References
For a short p approaches in agreement with / I / Likharev, K.K. and Yakobson, L.A., Proc. IEEE MAG-I1 (1975) 860
some experimental r e s u l t s 141. The s u p e r c u r r e n t i s
/2/ Jensen, H.H. and L i n d e l o f . R.E.. J . Low Temu.
a p e r i o d i c f u n c t i o n o f t , i t s time v a r i a t i o n a t t h e Phys.
23
(1976) 469 c e n t e r of WL i s shown i n f i g u r e 1 . 2.0 r Fig. 1 : V a r i a t i o n of t h e s u p e r c u r r e n t d e n s i t y J ( t ) , t h e p a i r d e n s i t y Ns ( t ) and J /Ns ( E r o p o r t i o n a l t o v e l o c i t y ) a t WL c e n t f e i n a Josephson c y c l e . a = 1.5, 9 = 0.1. D. The p a i r density.- Analogous t o t h e s u p e r c u r r e n t , t h e p a i r d e n s i t y can be w r i t t e n a sN ( t ) = No s i n $ + N2 (1 + q c o s $), t h e c o e f f i - c i e n t f o r t h i s . " c o s 6 term" i s :
% ( s i n a
-
a cos a ) 2 a - s i n 2 a/ 3 / Clark
