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Submitted on 1 Jan 1978
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COLLECTIVE MODES IN ARRAYS OF
SUPERCONDUCTING BRIDGES
S. Artemenko, A. Volkov, A. Zaitsev
To cite this version:
JOURNAL DE PHYSIQUE
Colloque
C6, suppldment au
no8, Tome 39, aotir 1978, page
C6-588
COLLECTIVE MODES
I N ARRAYS
OF SUPERCONDUCTING BRIDGES
S.N. Artemenko, A.F. Volkov and A.V. Z a i t s e v
I n s t i t u t e of Radioengineering and EZectronics, USSR Academy o f Sciences, USSR, Moscow, 103907, Marx Avenue, 1 8 .
R6sum6.- On a obtenu d e s 6 q u a t i o n s d s c r i v a n t l ' e f f e t Josephson dans d e s p o n t s supraconduc- t e u r s 1 une dimension. On montre que dans l e systPme d e ponts p r 6 s d e Tc, peuvent e x i s t e r des o s c i l l a t i o n s c o l l e c t i v e s dont l e s p e c t r e a une s t r u c t u r e de bande.
A b s t r a c t . - Equations d e s c r i b i n g t h e Josephson e f f e c t i n p r o x i m i t y - e f f e c t b r i d g e s were o b t a i n e d . I t was shown t h a t i n s e r i e s a r r a y s c o l l e c t i v e modes w i t h band spectrum can e x i s t near T
.
I t was shown e x p e r i m e n t a l l y / I / and theo- r e t i c a l l y / 2 , 3 / t h a t i n superconductors n e a r Tc weakly-damped c o l l e c t i v e o s c i l l a t i o n s of t h e su- p e r f l u i d v e l o c i t y p s / ~ and of t h e e l e c t r i c f i e l d E ( x , t ) =-V$(x,t) , w i t h a sound spectrum, can e x i s t . We s h a l l s e e t h a t t h e p o s s i b i l i t y of appearance of t h e s e modes and t h e l a r g e p e n e t r a t i o n ' l e n g t h of t h e f i e l d E i n t o a superconductor lead t o some in- t e r e s t i n g p r o p e r t i e s of t h e Josephson j u n c t i o n s . These e f f e c t s a r e e s p e c i a l l y important i n t h e case of weak l i n k s i n which t h e c u r r e n t d e n s i t y j de- pends only on one c o o r d i n a t e ( f o r example, a s i n t h e p r o x i m i t y - e f f e c t b r i d g e s 1 4 1 ) . R e s i s t a n c e of such b r i d g e s i s determined by t h e p e n e t r a t i o n l e n g t h of the f i e l d E i n t o superconductor. Let u s o b t a i n e q u a t i o n s d e s c r i b i n g t h e Josephson e f f e c t i n t h e p r o x i m i t y - e f f e c t b r i d g e made of t h e d i r t y superconductor ( T r < < l , ~ - i m p u r i t y c o l l i s i o n time). We s h a l l u s e t h e model of t h e b r i d g e considered i n / 5 / f o r t h e c a s e of g a p l e s s
superconductors, i .e.
,
we c o n s i d e r a t h i n super- conducting f i l m w i t h t h e c r i t i c a l temperature Tc(x) which depends on t h e c o o r d i n a t e along t h e b r i d g e :T ~ ( X ) = T: f o r I x l < d , T ~ ( x ) = T ~ > T: f o r 1x1 d.
Namely, a t ] x i < d , s i m i l a r l y t o / 5 / , t h e y reduce t o one l i n e a r Ginzburg-Landau e q u a t i o n f o r
Z\,
and a tI x I >
d , t h e i r s o l u t i o n i s K(x) = A t h R x + ~ ~ ) / ~ S ( ~ ~ e x p ~ ~ ( x > 7 . Matching t h e s o l u t i o n s f o r A(x) a t 1x1 = d , s i m i - l a r l y t o151,
we o b t a i n t h e e x p r e s s i o n f o r t h e c u r r e n t d e n s i t y j = j c s i n $ ( t ) + a E ( t ) (1)where E ( t ) = E(d, t )
.
(The f i e l d i n s i d e t h e b r i d g e i s independent of x),
$ ( t ) = 2x(d). To f i n d t h e r e l a t i o n between E ( t ) and $ ( t ) we have t o s o l v e t h e e q u a t i o n d e s c r i b i n g t h e process of conversion of t h e q u a s i p a r t i c l e c u r r e n t j n = a E ( x , t ) i n t o thesuperconducting one j =
$
(x) ps (x)where p= (112- g + e $ ( x , t ) , T -energy r e l a x a t i o n time. The r e l a t i o n between E ( x , t ) and one can be o b t a i n e d from t h e formula 3 s = e E ( x , t ) + Vp
a t
and from t h e c o n t i n u i t y e q u a t i o n f o r t h e t o t a l c u r r e n t j = jn + js. T h i s r e l a t i o n f o r t h e Fourier-transforms has t h e L e t parameter vo2 = (T-
<
) / ( T c-
T)>> 1 and formE ( X )
.
$
v,,
-
21WT
$1
-
W T ] - I (3)vod >S(T). Then t h e c r i t i c a l c u r r e n t of t h e b r i d g e w
3 6
-
1 I T A ~ ( X ) , *A (x)j = T j G L ~ ~ o ~ h ( 2 ~ o d / t ( ~ ) 7 1 is e x p o n e n t i a l l y
I t i s easy t o s o l v e Equations (2)
-
(3) and t o s m a l l i n comparison w i t h t h e d e p a i r i n g c u r r e n tj
GL
f i n d t h e sought r e l a t i o n between E ( t ) and $ ( t ) i fof t h e uniform f i l m / 5 / . I n t h e c o n s i d e r e d c a s e of
'xo t h e c h a r a c t e r i s t i c p e n e t r a t i o n l e n g t h k-1 of t h e
t h e o r d i n a r y superconductor t h e complex gap ~ ( x ) w
f i e l d E i s l a r g e enough : kW < ( T ) < < 1 . i s d e s c r i b e d by t h e g e n e r a l i z e d Ginzburg-Landau
(N)
, - ( j u / a ) i i 2 ( 2 v 0 < ( ~ )+
k-i) e q u a t i o n s w i t h t h e anomalous terms ( s e e , f o r exam- E,=4
a t
wp l e 1 6 1 ) . However, because of t h e smallness of d + k-:( ]-in)
-
2i6hroS(T)j c / j G L t h e s e e q u a t i o n s can be s i m p l i f i e d essentially. where Q =
?WT ,
k t = ( k * + ik9')2.
*(-iu + = - l )ITA 2 w
(-in + 1) (A/4TD).
Rather complicated form of (4) i s due t o t h e f a c t t h a t , provided w>>ri1 , A ~ / T , t h e p e n e t r a t i o n l e n g t h of E i n t o superconductors e s s e n t i a l l y depends on w . I n t h e low frequency l i m i t (w<<A2 /T,T -1 ) eE = 1 / 2 (%)[2(d
+
l E u - l . The q u a n t i t y:
E =k-I U=oi s t h e p e n e t r a t i o n l e n g t h of t h e low frequency f i e l d . Note a l s o t h a t t h e r e l a t i o n between Vw = 2(d
+
k;l)EW and ( a ( / a t ) d i f f e r s from t h e Josephson r e l a t i o n . But t h e l a t t e r i s v a l i d f o r time-averaged q u a n t i t i e s ( 2 e v ( t ) =m).
I t f o l - lows from (1) and (4) t h a t t h e h y s t e r e s i s may appear i n t h e I(V) curve / 7 / . Consider now a s e r i e s a r r a y of a l a r g e number N of microbridges ( i n expe- riment N may be a s l a r g e a s 2000 and t h e d i s t a n c e between a d j a c e n t b r i d g e s i s L=
111'1.
I n t h i s c a s e from ( 2 ) and ( 3 ) , we g e t f o r t h e f i e l d i n t h e n-th j u n c t i o n 1 E'"' ch(kwL) =+
( g ) w ( l - i ~ ) - ~ k ~ s h ( k ~ L ) +I
( ~ ( " 1 )+
82-1))+
( j W / c r ) i ~ ( l - i ~ ) - l Q - c h ( k w ~ ~ , (kWS(T)vo<<l) (5) provided d<rL, k i l . I f j = 0 and ( < < I , E ( ~ ) = W W - ( j c / a ) ( W ( n ) a s i t f o l l o w s from ( 1 ) . We s h a l l s e e k t h e s o l u t i o n of ( 5 ) , i n t r o d u c i n g t h e c o l l e c - t i v e c o o r d i n a t e s , $ + w 4 91
+
einqL, where q = 2mn/NL (m = 0 , 1 , 2 ...).
Then under t h e c o n d i t i o n s = - I < < ,, A ~ / T << U , w << A W (6) we g e t t h e d i s p e r s i o n r e l a t i o nA (w/wo)sin(b~/wo) = cos (w/wo)
-
cos (qL) (7)where h =wg/w: = ( l b ~ T j ~ / n L o A ) , w0 = (2DA) l I 2 / L . Consider two l i m i t i n g c a s e s . a) A < < 1 . I f Iw/wo
-
nnl > A , we have t h e sound-like spectrum of uniform superconductor u=(206) 1/2q2. I f lu/oo-nn1
< A,
t h e s p l i t t i n g of t h e spectrum a p p e a r s , and f o r b i d d e n frequency bands a r e formed. The magnitudes of t h e gaps a r e 6w= 2n wonk. b) A > > l . I n t h i s case t h e bands appear a l s o . I n t h e f i r s t band t h e spectrum h a s t h e form of t h e a c o u s t i c a l phonons branch i nc r y s t a l s
: w
= w s i n ( q L I 2 ) . I t i s important t o n o t e Jt h a t t h e mode i n t h e f i r s t band i s weaklyLdamped even i f t h e second c o n d i t i o n of (6) i s n o t f u l f i l - l e d , i . e . , even i f in an uniform superconductor t h e o s c i l l a t i o n s a r e heavily-damped. The modes i n o t h e r bands a r e weakly-damped o n l y under t h e condi- t i o n s ( 6 ) . The spectrum i s g i v e n by formula
w/wo = r n
+
( h ~ n ) - ' n-
( - l ) n c o ~ ( ~ L f l , n = 2,3...
( s e e F i g u r e ) .I n c o n t r a s t with t h e c a s e of uniform su- perconductor, t h e modes considered above cause p e c u l i a r i t i e s i n t h e impedance Z(w) of t h e conside-
red system. We c a l c u l a t e Z(w) on t h e grounds of (1) and ( 5 ) . The r e s u l t i s
Equating t h e e x p r e s s i o n i n s q u a r e b r a c k e t s t o zero, we o b t a i n t h e f r e q u e n c i e s of t h e c o l l e c t i v e modes
c
qL
T h e r e f o r e , t h e impedance p e c u l i a r i t i e s a r e determi- ned by t h e r e l a t i o n tg(w/2wo) = -A(w/wo). The lowest frequency i s marked on t h e f i g u r e by a c i r - c l e .
Besides, t h e p e c u l i a r i t i e s i n t h e I(V) cur- v e appear when a dc-voltage upon one j u n c t i o n
7
s a t i s f i e s t h e r e l a t i o n 2eV = ~ ( 2 ;+
li)w.
Calcula- t i n g t h e dependance I(V) a t l a r g e7,
we f i n d t h a t a c o r r e c t i o n j t o t h e ohmic c u r r e n t j =07 G l E t h
(L/21E)]-l, a s a f u n c t i o n of
v,
o s c i l y a t e s w i t h 4 2t h e p e r i o d nwo/e and w i t h t h e amplitude
-
j c kA2k"th(k'L/2U.We n o t e t h a t a n e q u a t i o n l i k e (5), d e s c r i - b i n g an i n t e r a c t i o n between j u n c t i o n s i s v a l i d not o n l y f o r considered one-dimensional a r r a y s , b u t a l s o f o r a r r a y s c o n s i s t i n g , f o r example, of v a r i a - b l e t h i c k n e s s microbridges. Thus, i f such two brid- g e s a r e s i t u a t e d a t a d i s t a n c e L<lE one from ano- t h e r , t h e n t h e y i n t e r a c t , and, i n p a r t i c u l a r , they may become spontaneously synchronised. The mode
locking c o n d i t i o n f o r p r o x i m i t y - e f f e c t b r i d g e s has t h e form
R e f e r e n c e s
/ I / C a r l s o n , R.A., Goldman, A.M. Phys. Rev. L e t t .
