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Z. P hys. B - Condensed M atter 85, 451-458 (1991)

ondensed

Zeitschrift a t t e r

for Physik

9 Springer-Verlag 1991

Q uasiparticle and oo pe r pa ir tunneling

in sm all capacitance Jo seph son junctions

f f ec t s o f the e l ec trom agnet i c env i ronm ent

G . F a l c i , V . B u b a n j a , a n d G e r d S c h 6 n

Department of A pplied Physics, Delft Universi ty of Technology, Lorentzweg 1, 2628 CJ Delft , The Netherlands

Received July 5, 1991

T h e t u n n e l i n g o f s in g l e e le c t r o n s i n s m a l l c a p a c i t a n c e

t u n n e l j u n c t i o n s i s i n f l u e n c e d b y c h a r g i n g e f f e c t s a n d b y

t h e f l u c t u a t io n s o f t h e e l e c r o m a g n e t ic e n v i r o n m e n t . W e

s t u d y t h e e f f e c t o f a n e x t e r n a l c i r c u i t w i t h a r b i t r a r y i m -

p e d a n c e o n t h e t u n n e l i n g o f q u a s i p a rt ic l e s a n d C o o p e r

p a i rs i n v o l ta g e d r i v e n J o s e p h s o n j u n c t i o n s . W e p r e s e n t

r e s u l t s a t f i n i t e t e m p e r a t u r e s a n d a l s o c o n s i d e r a n a c

d r i v e n s y s t e m .

1 . Int roduct i on

C h a r g i n g e f fe c t s i n s m a l l c a p a c i t a n c e t u n n e l j u n c t i o n s

h a v e r e c e n t ly a t t r a c t e d m u c h t h e o r e t ic a l a n d e x p e r im e n -

t a l i n t e r e s t . M o d e r n l i t h o g r a p h y a l l o w s t h e c o n t r o l l e d

f a b r i c a t i o n o f j u n c t i o n s w i t h c a p a c i t a n c e s i n t h e r a n g e

C < 1 0 1 5 F a n d l a r g e t u n n e l i n g r e s is t an c e Rt>;>R

= h / 4 e 2 ~ 6 . 4 5 k f ~ ) . I n t h e s e s y s t e m s t h e e n e r g y d i f f e r -

e n c e a s s o c i a t e d w i t h a s i n g l e e l e c t r o n t u n n e l i n g ( S E T )

e v e n t , w h i c h i s o f o r d e r E c - e 2 / 2 C , i s l a r g e e n o u g h t o

m a n i f e s t i ts e l f a t t e m p e r a t u r e s b e l o w T ~ 1 K . A t l o w v o lt -

a g e s [ V [ < e / 2 C t u n n e l i n g s h o u l d b e s u p p r e s s e d ( ' C o u -

l o m b b l o c k a d e ' ) , a n d f o r l a rg e v o l ta g e s th e I - V c h a r -

a c t e r is t ic s h o w s a n o f f s e t b y V = _+ e / 2 C ( ' C o u l o m b g a p ' )

[ 1 , 2 ] . T h e s e e f f e c t s h a v e b e e n c l e a r l y d e m o n s t r a t e d i n

n o r m a l c o n d u c t i n g , m u l t i - j u n c t i o n d e v i c e s [ 3 ] . I n e x p e r -

i m e n t s w i t h a s i n g le j u n c t i o n t h e e f f e c ts c a n o n l y b e o b -

s e r v e d i f t h e j u n c t i o n i s s u f f i c i e n t ly d e c o u p l e d f r o m t h e

e x t e r n a l c i r c u i t [4 ]. I n s u p e r c o n d u c t i n g j u n c t i o n s , i f t h e

J o s e p h s o n e n e r g y E j i s n o t t o o l a r g e, o n e m a y a l s o e x p e c t

a s u p p r e s s io n o f C o o p e r p a i r t u n n e l i n g a t l o w v o l ta g e s .

T h i s h a s b e e n r e c e n t l y o b s e r v e d [ 5] i n a s i n gl e j u n c t i o n ,

w h e r e a s t h e e x p e r i m e n t a l d a t a o n m u l t i- j u n c t io n s y s t e m s

* Present address: Istituto di F isica, Facolt/t di Ingegne ria, Uni-

versitfi di C atania, viale A. D oria 6, 1-95129 Catania, Italy

** New address : Institut ft ir The oretisch e Festk6rperphysik,

Universi tfi t Karlsruhe, W -7500 Karlsruhe, F ederal R epub lic of

Germany

s h o w m u c h m o r e s t r u c t u r e a n d s t i l l a w a i t a c o m p l e t e

t heo re t i ca l de scr i p t i on [6, 7 ].

T h e p r o p e r t i e s o f s m a l l t u n n e l ju n c t i o n s a r e i n f l u e n c e d

b y t h e e x t e r n a l c ir c u i t. T h e C o u l o m b b l o c k a d e is e f f e c t iv e

o n l y i f t h e c h a r g e r e l a x a t i o n t h r o u g h t h e c i r c u i t is s lo w .

T h e r m a l a n d q u a n t u m f l u c t u a t i o n s o f t h e e l e c tr o m a g n e t i c

e n v i r o n m e n t c a n a c t i v a t e t h e c h a r g e t r a n s f e r a c r o s s t h e

j u n c t i o n . T h i s w e a k e n s t h e C o u l o m b b l o c k a d e . T h e i n -

f l u e n c e o f th e e n v i r o n m e n t , w h i c h i s e x c i t e d b y t h e t u n -

n e l i n g c u r r e n t , b a c k o n t o t h e t u n n e l i n g i t s e l f w a s f i r s t

d i s c us s e d b y N a z a r o v [ 8]. T h e I - V c h a r a c te r i st ic s o f

n o r m a l t u n n e l j u n c t i o n s , e i t h e r c u r r e n t b i a s e d ( I x ) w i t h

a para l l e l s hun t r es i s t o r R~ [2 , 9 ], o r vo l t ag e b i ase d (Vx)

wi t h a se r i es r es i s t o r have been der i ved [10 ] (bo t h c i r cu i t s

a r e e q u i v a l e n t i f o n e r e p l a c e s V b y I x / R s ) . I n n o r m a l

j u n c t i o n s t h e l e a d i n g l o w t e m p e r a t u r e a n d v o l t a g e b e -

ha v i o u r i s I oc V) +l /2~s a t T = 0 a n d I, oc

Vx T 1/2~ t

f in i te te m p e r a t u r e s [9 ]. H e r e a s = R q / R , is th e d i m e n -

s i o n le s s c o n d u c t a n c e o f th e r e s i st o r . I n J o s e p h s o n j u n c -

t i o n s t h e i n f l u e n c e o f s h u n t o r s e r ie s r e s i s to r s o n t h e s u -

p e r c u r r e n t h a s b e e n a n a l y z e d . A t l o w v o l t a g e s i t i s

Is OC V -1 2/~

a n d I, oc

VT -2+2/~

f o r T = 0 a n d f i ni te

t e m p e r a t u r e s , r e s p e c t iv e l y [ 2, 1 1 ] . F o r w e a k c q a r e s o n a n t

s t r u c t u r e i n t h e s u p e r c u r r e n t a t V = e / C h a s b e e n p r e -

d i c t e d [ 1 2] . F r o m t h e s e r e s u l ts o n e s e es t h a t t h e C o u l o m b

b l oc kad e i s on l y e f f ec t i ve i f the se r i es r es i s t ance R s i s

s u b s t a n t i a l l y l a r g e r t h a n Rq, w h i c h c a n b e a c h i e v e d in a n

e x p e r i m e n t o n l y w i t h s o m e e f f o r t [ 4 ] .

T h e e f f e c t o f a n a r b i t r a r y l i n e a r c i r c u i t c a n b e a c -

c o u n t e d f o r b y t h e c i r c u i t i m p e d a n c e Z ( co ), a s s e e n b y

t h e j u n c t i o n [ 13 , 1 0 ] . W i t h t h i s i n m i n d w e c a n d e s c r i b e

t h e j u n c t i o n p l u s e n v i r o n m e n t i n a c o m p a c t f o r m b y a n

e f f e c t iv e a c t i o n , c o m b i n i n g t h e a c t i o n d e s c r i b i n g C o o p e r

p a i r a n d q u a s i p a r t i c l e t u n n e l i n g [ 1 4 ] , a n d t h e a c t i o n o f

t h e e x t e r n a l i m p e d a n c e [ 1 3 ] . A n e x p a n s i o n i n t h e w e a k

t u n n e l i n g l i m i t 1 / R ~ < I / Z o ) ~ O ) r e p r o d u c e s t h e e x -

p r e s s i o n f o r t h e t u n n e l c u r r e n t d e r i v e d i n [ 10 ] f o r n o r m a l

j u n c t i o n s [ 1 5, i 6 ] . I t c r u c i a l l y d e p e n d s o n t h e p r o b a b i l i t y

f o r i n e l a s ti c tu n n e l i n g p r o c e s se s P ( E ) w h i c h d e s c r i b e s t h e

f a c t t h a t t h e e l e c t r o m a g n e t i c e n v i r o n m e n t h a s t o b e e x -

c i t e d f o r t u n n e l i n g t o t a k e p l a c e . I t h a s b e e n s h o w n b e f o r e



452

[17] tha t in the weak tunnel ing l imi t the quas ipar t i c le

c u r r e n t i n J o s e p h s o n ju n c t i o n s a n d t h e s u p e rc u r r e n t [ 12 ]

a l so depe nd o n the sam e func t ion P (E) . In th i s a r t ic le

we wi ll s tudy fu r ther the e f fec t o f f in i te t em pera tu r es :

b e s i d e t h e e x p e c t e d o v e ra l l w e a k e n i n g o f t h e c h a rg i n g

effec t s , the ac t iva t ion by the env i ron me n t l ead s to a les s

o b v i o u s m o d i f i c a t io n o f t h e q u a si p a r ti c l e c u r r e n t a s u b g a p

v o l t a g e s . F u r t h e r w e e x t e n d o u r a n a l y s i s t o s t u d y t h e

e f f e ct o f a n a c d r iv e . F o r t h e s a k e o f c o m p l e t e n e s s w e

re p e a t t h e fo rm u l a t i o n o f th e p ro b l e m g i v en b e fo re i n

[17] in the fol lowing sect ion.

2 M o d e l a n d m e t h o d s

W e cons ider the c i rcu i t sho wn in F ig . 1 . Bo th on the

c l as s ic a l a n d t h e q u a n t u m m e c h a n i c a l l e v e l a n a rb i t r a ry

e n v i ro n m e n t c o m p o s e d o f li n e a r e l e m e n ts ( e . g . a t r an s -

m i s s io n l i n e ) c a n b e a c c o u n t e d fo r b y u s i n g a s u i t a b l e

i m p e d a n c e Z (o ) ) . Ea c h c i r c u i t e l e m e n t i s c h a ra c t e r i z e d

by a phase wh ich i s re la ted to the re la t ive vo l tage d rop

b y

~(o~/~t=2eV~

we pu t h = 1 ) . The ind ices i= t ; z ; x

re fer to the tunnel junc t ion ( t ) , s e r ies impedance (z ) , and

vo l tage source (x ) , respec t ive ly . In the imag inary - t ime

p a t h - i n te g ra l f o rm a l i s m t h e e f f e c ti v e a c t i o n o f t h e t u n n e l

j u n c t i o n p l u s e x t e rn a l i m p e d a n c e i s

s r ) ; r ) ] = s , [ r r ) ] + s r ) ] .

(1)

The ac t ion o f the tunnel junc t io n i s g iven by [14]

S t [ ~ b t (r ) ] = ~ d r I C ( 1 e ~ t t t ) 2 - E j c o s q ~ t ( r ) l

B

+ ~ d r ~ d r o c t ( r - U )

o s

( 2 )

I t d e sc r i b e s s u p e rc o n d u c t i n g a n d n o rm a l j u n c t i o n s : t h e

f i r s t t e rm accoun ts fo r the charg ing energy , the second

fo r t h e J o s e p h s o n c o u p l i n g ( i f p r e s e n t ) , a n d t h e e t - t e rm

for the tunnel ing o f quas ipar t i c les in supercon duct ing

j u n c t i o n s (o r s i n g l e e l e c t ro n s i n n o rm a l j u n c t i o n s ) . Th e

k e rn e l e , ( r ) s c a l e s w i t h t h e d i m e n s i o n l e s s n o rm a l s t a t e

t u n n e li n g c o n d u c t a n c e ~ , Rq/Rt, bu t i t s func t iona l fo rm

depends on the gap , re f lec t ing the co rrespond ing s t ruc-

t u r e s i n t h e q u a s i p a r ti c l e I - V c h a ra c t e r is t ic s . F o r fu r th e r

de ta i l s we re fer to [2 ] o r [14 ] . The ac t ion descr ib ing the

ex terna l c i rcu i t , a f t e r F our ie r t ra ns fo rm at ion , i s [ 13 ]

V

Fig 1. The e quivalent circuit for a v oltage biased tunnel junction,

the external impedance simulating a general electromagnetic envi-

ronment

1 I o l z _ l

s 1 6 2

] ) I C z ( o J ) [ 2 . ( 3 )

The vo l tage source f ixes ~b x / ~ t = 2e V . The phases q~,

and ~bz a r e s u b j e c t t o b o t h t h e rm a l a n d q u a n t u m f l uc -

tuat ions with the c ons train t q~, + ~bz + q~x = 0 i m p o s e d b y

the c i rcu i t . L arge f luc tua t ions o f q~z a re poss ib le i f the

impedance in (3 ) i s l a rge enough and imply l a rge f luc-

t u a t i o n s o f ~b t, l e a d in g t o t h e u s u a l C o u l o m b b l o c k a d e

p ic tu re . The re lev an t averages o f observab les re fe r r ing to

t h e j u n c t i o n a r e c a l c u l a te d a s

A

r

= Z o S D e ,

( r ) ( r )

• ( (~ t+(~ +~ x)A (~bt ) exp { - S[q~t; ~ ] }

( 4 )

whe re Z 0 i s the par t i t ion func t ion . Fo r in s tance , the cu r-

r e n t f l o w i n g t h ro u g h t h e j u n c t i o n c a n b e e x p re s s e d a s

( I ( r ) = - 2 e E s (s in ~b~(r )}

( r G ( r ) )

- 2 e j d r a t ( r - r ) sin 2

0 ( 5 )

The f i r s t t e rm i s the supercu rren t , the secon d i s the quas i -

par t i c le ( s ing le e lec t ron ) cu rren t fo r a superconduct ing

(n o rm a l ) j u n c t i o n .

I f th e t u n n e li n g c o n d u c t a n c e fo r q u a s ip a r t ic l e s 1 / R ,

a n d t h e s t r e n g t h fo r C o o p e r p a i r t u n n e l i n g E j a r e w e a k

w e c a n p ro c e e d p e r t u rb a t i v e l y i n t he s e p a r a m e t e r s [ 8 -1 2 ,

15 , 17 ] . S ince the junc t ion coup les p red om inan t ly to the

l o w f r e q u e n c y m o d e s o f th e e l e c t ro m a g n e t i c e n v i ro n m e n t ,

and in mos t exper imen ts Z(o~) i s main ly res i s t ive fo r

s m a l l ~ , t h e r e q u i r e d c o n d i t i o n s a r e 1 / R r 1 / Z (0 ) and

E j / E c ~ R q / Z

(0 ). Fo r l a te r u se we def ine ~s

Rq / g 0 ) .

In the cons idered l imi t the ac t ion reduces to

S ~ S o [ r d r 5 2-e t~r / +S~[q~x+~b~] (6 )

0

a n d t h e a v e ra g e s , t o b e e v a l u a t e d w i t h t h e q u a d ra t i c

S o [~bt], can be expre ssed explici t ly .

In (6 ) we have e l imina ted the exp l ic i t dependence on

~b~ in fa vo ur of the drive q~x. This pro ced ure is s l ight ly

d i f fe ren t than the one used in [17 ] and i s m ore conv en ien t

i f we a l low fo r a genera l d r ive V~(t). The averages ca l -

c u l a t e d i n t h is w a y c a n b e f a c t o r i z e d i n t o a n a v e ra g e o v e r

t h e e q u i li b r i u m f l u c t u a t i o n s a n d a t e rm d e p e n d i n g o n t h e

d r ive . The l a t t e r tu rns ou t to depend on the c las s ica l

voltage at the junction V on ly . For in s tance

( si n ~ t ( r ) 2 ~ t ( r ' ) )

I lo~, D ( ~ ) (ei o j~ _ ei o jr, ) ~x (~o)1

= s i n - ~ 8 f l e 2 Z ( _ il o ) l )

• 1 - c ~

4 f l D (~

=sin I dseV(s) l exp{-}([c~ t (r) - -r



o)~ I o ) l z - 1

w h e r e D ( c o ) = ~ E ~ + ~ e S - e 2

- i l co

[ ) i s t he equ i l i b -

r i u m p r o p a g a t o r i n F o u r i e r r e p r e s e n t a ti o n . T h e s u b s c r ip t

e q r e f e r s t o e q u i l i b r i u m a v e r a g e s , o b t a i n e d b y t h e a c t i o n

(6 ) wi t h q5 ~ = 0 . S i nce t he c i r cu i t d escr i be d by t he e f f ec t i ve

a c t i o n ( 6 ) is l i n e a r , V i s a l s o t h e q u a n t u m m e c h a n i c a l

a v e r a g e ( V ~ >. N o t i c e t h a t f r o m t h e f o l l o w i n g a n a l y s i s w e

w i l l e x p r e s s t h e c u r r e n t a s a f u n c t i o n o f t h e v o l t a g e

at

the junction .

In con t r as t i n [10 ] t he d r i ve V~ app ear s as

t h e i n d e p e n d e n t v a r i a b l e i n t h e c h a r a c t e r i s t i c s . T h i s i s

v a l i d o n l y i f V ~ i s c o n s t a n t a n d i n t h e w e a k t u n n e l i n g

l i m i t : i n t h i s c a s e i n d e e d V ~ V . T h e f o r m u l a t i o n r e-

p o r t e d h e r e p r o v i d e s a d e s c r i p t i o n v a l i d i n m o r e g e n e r a l

c i r c u i ts a n d a p p r o x i m a t i o n s c h e m e s ( s e e S e c t. 6 ) a n d i n

a w i d e r r a n g e o f p a r a m e t e r s a l l o w i ng u s t o s o f t e n t h e

r e q u i r e m e n t 1 / R

t ~ 1 /Z (0) .

T h e f i r s t o r d e r c o n t r i b u t i o n t o t h e q u a s i p a r t i c l e t u n -

n e l i n g c u r r e n t r e d u c e s t o

< I q p ( r ) > = - 2 e

~ d r ' ~ , ( r - z ' ) s i n

d s e V ( s )

0

• - 1 < [ ( ]~ t ( T ) - - q ~ t ( c ' ) ] 2 > e q }

(7)

a n d t h e s u p e r c u r r e n t, w h i c h c o n t r i b u t e s i n s e c o n d o r d e r

i n E j , t o

< I , ( r ) >

= e E ) ~

d r ' s i n

d s 2 e V ( s )

0

• - 89 ( r ) - qS, ( r ' ) ]2 >e q}. (8)

T h e e q u i l i b r i u m a v e r a g e s c a n b e c a l c u l a t e d b y t h e f l u c -

t u a t i o n - d i s s i p a t i o n t h e o r e m a n d a r e c o m p l e t e l y d e t e r -

m i n e d b y t h e i m p e d a n c e s e e n b y t h e j u n c t i o n

z ~ ( o ) ) = [ i o ) c + z ( c o ) - ] - ~ .

A f t e r a n a l y t i c c o n t i n u a t i o n t o r e a l t i m e s r - - * i t t h e s e

e x p r e ss i o ns b e c o m e

• I m { ~ , > ( t - t ' ) e K ( ' - ' ') }

(9)

a n d

( I ( t ) ) = - 2 e E ~ i dt

--03

• I i d s 2 e V ( s ) l I m e g K (t -c )

w h e r e

(10)

K t ) = 1 < [ ~ )t t ) - ~ t ( 0 ) ] ( ~ t ( O ) > e q

e 2 ~ R e Z c ( o ) )

d o )

7 r o )

o

I

co th ~ - (cos co t - 1) - i s in o) t

(11)

453

a n d t h e d e t a i le d f o r m o f t h e k e r n e l e t > ( t ) i s g i ven

in [14].

3 . d c t u n n e l c u r r e n t a t T = 0

F o r a d c d r i v e t h e a v e r a g e q u a s i p a r t i c le c u r r e n t i s r e a d i l y

ob t a i n ed [ 17 ]

l + o o + o r

I qp -- eR ~ d E ~ d E ' N ( E ) N ( E ' )

--co 03

• f ( E ) ] P ( E - E +eV

- [ 1 - f ( E ) l f ( E ' ) P ( E ' - - E - - e V ) } ( 1 2 )

T h e q u a s i p a r t i c l e c u r r e n t t u r n s o u t t o b e a t r a n s p a r e n t

e x t e n s io n o f th e c o r r e s p o n d i n g e x p r e s si o n i n a n o r m a l

j u n c t i o n [ 10 ]. I t i n c lu d e s t h e r e d u c e d d e n s i ti e s o f q u a s i -

par t i c l e s t a t es

N ( E )

a n d

N ( E ' )

i n t he t wo e l ec t rodes .

The supercu r ren t i s [12 ]

+ c o

I s = - Z e s ~ ~ d t si n ( 2 e V t) I m e x p [ 4 K ( t )]

(13)

o

T h e q u a n t u m e f f e c t s a r e c o n t a i n e d i n t h e f u n c t i o n

P ( E ) i n t r o d u c e d i n [ 1 0 ]

P ( E ) = ( 1 / 2 r c ) ~ d t e x p [ K ( t ) + i E t ] .

(14)

co

T h e f u n c t i o n P ( E ) c a n b e i n t e r p r e t e d a s t h e p r o b a b i l i t y

t h a t t h e e n e r g y E i s e x c h a n g e d w i t h t h e e n v i r o n m e n t . I t

i s t h e c e n t r a l q u a n t i t y t o b e c a l c u l a t e d . I t s a t is f ie s d e t a i l e d

b al an c e P ( - E ) = e - # e p

(E).

In t he c l ass i ca l l i mi t i t re -

d u c e s t o a 6 ( E ) - f u n c t i o n , a n d w e r e c o v e r t h e u s u a l e x -

p r e s s i o n f o r t h e t u n n e l i n g c u r r e n t , n a m e l y f o r n o r m a l

j u n c t i o n s a l i n e a r c h a r a c t e r i s t ic a n d f o r t h e q u a s i - p a r t i c l e

c u r r e n t t h e G i a e v e r - W h e r t h a m e r [ 1 7] f u n c t i o n o r a p p r o -

p r i a t e g e n e r a l i z a t i o n s t o n o n - i d e a l c a s e s .

I n g e n e r a l t h e q u a s i p a r t i c l e ( n o r m a l ) c u r r e n t c a n b e

e x p r e s s e d a s

I qp ( v )

co

= ~ d E 1 - e x p ( - B e V )

P ( e V _ E ) i c ~ ( E / e )

(15 )

1 - exp ( - f i E )

- - o o

w h e r e I c l ( V ) i s th e c h a r a c t e r i s t ic s f o r a p e r f e c t l y v o l t a g e

b i a s e d s y s t e m . T h e s u p e r c u r r e n t i s [ 1 2 ]

I s ( V ) = ~ e E Z { P ( 2 e V ) - - P

( - 2 e V )} ( 1 6)

w h e r e t h e f u n c t i o n P ' ( E ) i s t h e F o u r i e r t r a n s f o r m o f

e x p [ 4 K ( t ) ] . I t is o b t a i n e d f r o m P ( E ) b y t h e s u b s t i t u t i o n

Zc--* Zb = Zc / 4 .

W e d e n o t e b y K o ( t ) t h e c o r r e l a t io n f u n c t i o n

K ( t )

a t

T = 0 a n d i n t r o d u c e t h e c o r r e s p o n d i n g P 0 ( E ) . I n a l l c a s e s

o f i n t e r e s t e x p [ Ko ( t ) ] d e c a y s a l g e b r a i c a l l y a n d w e e x p e c t

n o n - a n a l y t i c b e h a v i o u r o f P o ( E ) f o r E = 0 . S i n c e t h i s

m a k e s t h e n u m e r i c a l i n t e g r a ti o n s o f ( 1 t ) a n d ( 1 4 ) c u m -



454

b e r s o m e , w e c a l c u l a t e d i r e c t l y P 0 ( E ) , o m i t t i n g t h e c a l -

c u l a t i o n o f K o ( t ) . F o l l o w i n g M i n n h a g e n [ 1 9] w e c a n d e -

r i v e a n i n t e g r a l e q u a t i o n f o r P 0 ( E )

z R e Z c ( E - o ) ) P o(o ))

(17)

E P o ( E ) = [ . d o ) 2 R q

0

w h i c h w e o b t a i n f r o m th e F o u r i e r t r a n s f o r m o f ( d / d t )

e x p [ K0 ( t ) ] c a l c u l a t e d f r o m ( 1 1 ) a n d ( 1 4 )

co

d e x p [ K o ( t ) ] = j. d o ) e _ i ~ o t i o ) p ( o ) )

d t

co

co

i e x p [ K 0 ( t ) ] y d o ) R e Z c ( o ) ) - i~ ,~

2 R q

0

a n d b y o b s e r v i n g t h a t

P o ( E ) = 0

f o r E < 0. T h e l a s t

p r o p e r t y c o r r e s p o n d s t o t h e f a c t t h a t a t T = 0 th e e n v i-

r o n m e n t i s i n it s g r o u n d s t a t e a n d c a n o n l y b e e x c i t ed b y

a t u n n e l i n g p r o c e s s .

E q u a t i o n ( 1 7 ) g i v e s

P o ( E )

u p t o a m u l t i p l i c a t i v e

c o n s t a n t , w h i c h w e f ix b y t h e n o r m a l i z a t i o n c o n d i -

t i o n . O t h e r s u m r u l e s c a n b e d e r i v e d f r o m c o n s i d e r i n g

d e x p [ K o ( t ) ] /d t ~

f o r t = 0 , e .g .

d o )

d E E P o ( E ) = 2 e 2 ~ ~

R e Z c ( o ) )

0 0

( 1 8 )

T h e 1 .h .s. i s t h e a v e r a g e e n e r g y a b s o r b e d b y t h e e n v i r o n -

m e n t a n d c o r r e s p o n d s t o t h e o f f s e t o f t h e I q p (~) ( V x) c u r v e s

f o r l a r g e v o l t a g e s. F o r a p u r e l y r e s is t iv e e n v i r o n m e n t t h e

r . h . s , g ives

E c.

F r o m t h e o t h e r s u m r u l e s a n d ( 1 7 ) t h e

d e c a y l a w o f P o ( E ) f o r l a r g e E c a n b e i n f e r r e d , n a m e l y

P o ( E ) ~ E - 1R e [ Z c ( E ) / 2

Rq].

E q u a t i o n ( 1 8 ) al s o i m p li e s

t h a t P0 ( E ) ~ E - ~+ ~ /2~ , fo r sm a l l E , w her e a s =

R u / Z ( O )

a n d t h e a f o r e m e n t i o n e d l i m i ti n g b e h a v i o u r s f o r s m a ll V~

o f t h e I - V c h a r a c t e r is t i c s c a n b e d e ri v e d . T h e r e s u l t in g

P o ( E ) f o r d i f f e r e n t v a l u e s o f t h e e n v i r o n m e n t a l i m p e -

d a n c e i s s h o w n i n F i g . 2 . W i t h d e c r e a s i n g

a , = R q / Z ( O )

e2p

2C

2

0 . 5 . 5

1 2 2 . 5

2CE

e2

Fig. 2. The normalized probability of inelastic tunneling P (E) in

units of

1/Ecis

shown for T = 0 and cq = 50, 5, l , 3, 0.2, 0.05, and

0.0125 from left to right, respec tively. Acco rding to (16) this is also

the T= 0 plot of

eI,/2 ~ CE~

vs.

C Vie

for values o f cq which are

4 times larger than those q uoted above

2 b , , . . . . . . . , j / /

l e

0 1 2 ~ / ~

5 ;

2 C V

e

Fig. 3. The quasiparticle I - V characteristic at T = 0 is plotted for

e~=h/(4eZR~)=oo,

10, 5, 1, 0.3, 0.05, 0.0125 and 0, from left to

right, respectively. We choose A = 2

E c.

The asymptotes for high

voltages for the Giaever curve I~ ( c ~ oo) and the other curves are

indicated by d ashed lines. The inset sh ows the tunneling current

I, (V) in a normal junction for the sa me set of cq

t h e p e a k i n P0 ( E ) m o v e s f r o m E = 0 t o E =

E c .

T h e f u n c -

t i o n a l f o r m o f P o ( E ) d i f f er s q u a l i t a ti v e l y f o r a m< 1 / 4 ,

f o r 1 / 4 < a , < 1 / 2 , a n d f o r 1 / 2 < c q . F o r a p u re ly

O h m i c e n v i r o n m e n t

P o ( E ) = O ( E )

f o r

R q / R s ~ O O ,

a n d

P o ( E ) = f i ( E - - E c )

f o r

R q / R ~ O .

I n F i g . 3 t h e r e s u l t i n g q u a s i p a r t i c l e c u r r e n t i s p l o t t e d

[ 1 7 ] . Q u a l i t a t i v e l y d i f f e r e n t r e s u l t s e m e r g e f o r a s < 1 / 2

a n d ~ s > 1 / 2 . T h e c u r r e n t I qp ( V ) i s m o r e s t r u c t u r e d t h a n

t h e n o r m a l j u n c t i o n r e s u l t I , ( V ) s h o w n i n t h e i n se t . I n

f a c t , I q p i s r e m i n i s c e n t o f t h e d e r i v a t i v e

d I , / d V .

P r o -

n o u n c e d f e a t u r e s o f t h e c h a r g i n g e f f e c ts a r e a p p a r e n t e v e n

f o r r e l a t i v el y s m a l l v a lu e s o f Z ( 0 ) . F o r i n s t a n c e t h e j u m p

a t e

V = 2 A

in the c la s s ica l cha rac te r i s t i c /~ j i s sm eared ,

a n d i n i ts v i c i n i t y I qp i s l o w e r t h a n I ~t . F r o m t h e e x p a n s i o n

o f P ( E ) m e n t i o n e d a b o v e w e f i n d a t T = 0 f o r e V l a r g e r

b u t c l o s e t o 2 A

eV

I q p V ) = ~

d E P E ) l c l e ~ f - E )

0

1

e R t r ( 1 / a , ) ~ ( e V - 2 A )

(19)

w h e r e a s I qp v a n i s h e s f o r e V < 2 A . F o r l a r g e v o l t a g e s

Iq p ( V ) t e n d s t o f o l l o w a s h i f te d G i a e v e r - W h e r t h a m e r

c u r v e , s h o w i n g t h e C o u l o m b g a p _+ e / 2 C in t h e s a m e

w a y a s t h e n o r m a l s t a te I n ( V ) .

T h e e n v i r o n m e n t a l s o i n f l u e n c e s t h e s u p e r c u r r e n t

I , ( V ) ( 1 6 ) . T h e s u p e r c u r r e n t h a s a p e a k w h i c h s h i f ts f r o m

V = 0 f o r ~ s- - o o t o V = e / C f o r c % ~ 0 , a n d t h r e e r eg i m e s

c a n b e d i st in g u is h e d, n a m e l y a , < 1 ,1 < c % < 2 ,

a n d 2 < a s . F o r l a r g e a s w e r e c o v e r t h e c l a s si c a l r e s u l t

I~ = lrE~ {~s Vz

[1 + (Jr

Vx

C / e c ~ s 2 ] } - i

f o r l a r g e V . , w h e r e -

a s f o r s m a l l v o l t a g e s t h e p e r t u r b a t i v e a p p r o a c h f a i l s t o

d e s c r i b e a d e q u a t e l y t h e c l a s s i c a l s u p e r c o n d u c t i n g s t a t e .

F o r s m a l l % < 1 t h e C o o p e r p a i r t u n n e l i n g i s b l o c k e d a t

l o w v o l t a g e s , f o r v e r y s m a l l c% t h e s u p e r c u r r e n t s h o w s

t h e s h a r p r e s o n a n c e a t

e / C



4 . d c t u n n e l c u r r e n t a t f i n i t e t e m p e r a t u r e s

A t f in i te t e m p e r a t u r e s i t i s c o n v e n i e n t t o s p l it

K ( t ) = K o ( t ) + K ~ ( t ) w h e r e

K l t) = e ~ - ~ d ( ~ e c(c~

• c o t h ~ - I ( c o s c o t - 1 ) (2 0)

a n d t o d e f i n e t h e c o r r e s p o n d i n g f u n c t i o n P 1 ( E ) , w h i c h

c a n b e e v a l u a t e d b y d i r e c t i n te g r a t i o n , P ( E ) b e i n g t h e

c o n v o l u t i o n o f P 0 ( E ) a n d P~ ( E ) . T h e r e s u l t s o n t h e I - V

c h a r a c t e r i s ti c s a r e s h o w n i n F i g . 4 a n d 5 . A s i n n o r m a l

t u n n e l j u n c t i o n s [ 20 ], t h e C o u l o m b b l o c k a d e g e ts w e a k -

e n e d b y fi n it e t e m p e r a t u r e s. F u r t h e r m o r e , t h e n o n - a n a -

l y t ic b e h a v i o u r i n Iq p a n d I~ is r e m o v e d a t a r b i t r a r y s m a l l

t e m p e r a t u r e s ( c o m p a r e t h e T = 0 r e su l t s in F i g . 3 a n d 2 ,

a n d t h e f i n i t e T r e s u l t s i n F i g s . 4 a n d 5 ) .

I n s e v e r a l l im i t s a n a l y t i c e x p r e s s i o n s o f K ~ ( t ) c a n b e

o b t a i n e d . T h e d o m i n a n t c o n t r i b u t i o n i n ( 20 ) a ri se s fr o m

t h e f re q u e n c i e s ~ < T . T h e i m p e d a n c e Z c ( c n ) d e c a y s

1

s u b s t a n t i a l l y o n a f r e q u e n c y s ca l e g i v e n b y R ~ ( w h e n

t h e e n v i r o n m e n t i s a s i m p l e r e s i s t o r ) . H e n c e f o r

T ~ l / ( R , C ) = 4 o c ~ E c / r C w e c a n u s e Z c ( O ) i n t h e i n t e -

g r a t i o n , o b t a i n i n g

K i t ) , ~ _ 1 in s inh rc T i t ] ) 21)

a s ~ T I t I

F o r l o n g t i m e s ( 2 1 ) r e d u c e s t o t h e ' d i f f u s i v e ' f o r m

K 1 ( t) ,. ~ - - D ] t I w h e r e D = r c T / ( 2 a ~ ) , w h e r e a s f o r s h o r t

u , , j . , / .

4 . ' . . . . ' . . . . ' - -

2CRt lqp '~ ~ - ~ -

2 I / '

4 . / /

3 . 8 4 4 . 2 4 . 4

2

0 - ~ 3 4

r ~ 4 ;

2 C V

Fig. 4. The quasiparticle current is plotted for finite temperatures

T / E c = 0.25 (corresponding to T~ 20 0 m K i f C= 10 i s F) . For ref-

erence the G iaever result at the sam e T and a shifted one are shown

by dashed l ines . Here A =e2/C and c~,=5, 1.2, 0.8, 0.3 and 0.05

from left to right, respectively. The b otto m right inset shows a blow

up o f the subg ap region , here Iqp (in logarithmic scale) increases

with decreasing ~,. The top left inset shows the results at lower

temperature T / E c = 0.025 (T ~ 20 mK i f C = 10-~5 F) in the region

V ~ 2 A . At this temperature the mos t pronounced ef fect com pared

to T = 0 is the vanishing of the nonanalici ties

455

2gCE j 2

1.5 4

3

0.5

0

0 1 2 3 4

C V

e

Fig. 5. The supercurrent for finite temperatures T/Ec=0.25

(T~ 20 0 m K i f C = 10 - i s F) and c%= 5, 1 .2, 0 .8, 0 .3 and 0.05 f rom

left to right , respectively, with the w eakening of the C oulom b block-

ade. The inset show s the results at lower tempe rature T / E c = 0.025

T ~ 20 m K i f C = 10 -15 F )

t i m e s i t is K 1 ( t ) ~ - 1 / ( 1 2 a , ) ( g T / ) 2. F o r a , ~ > 1 t h e d i f -

f u s iv e f o r m i s s u ff i ci e n t t o d e t e r m i n e / ' 1 ( E ) , w h e r e a s i n

t h e o p p o s i t e li m i t a s ~ 0 . 1 t h e s h o r t t im e e x p a n s i o n c a n

b e u s e d .

I f T >> 1/ (R s C ) t h e i m p e d a n c e Z c ( o ) ) p r o v i d e s a n e f -

f e c t iv e h i g h f r e q u e n c y c u t o f f l e a d in g t o

2 D l t [ a r c t g ( 4 a ~ E c ] t l )

+ 8 ~ T 2 1 n [ l + ( 4 a ~ E c ] t l ) 2 ] . (2 2)

I n t h e l o n g t i m e s l im i t a l so t h i s f o r m r e d u c e s t o t h e d i f -

f u s i v e f o r m , w h i c h i s s u f f i c i e n t t o d e t e r m i n e P 1 ( E ) i f

T ~ E c a 2, w h e r e a s f o r s h o r t t i m e s w e g e t

7 E c T t2 (23)

K i ( t ) ~ 2 z c

w h i c h c a n b e u s e d t o c a l c u l a t e P ( E ) i f T ~ > E c a ~ .

L e t u s c o n c e n t r a t e o n t h e w i d t h o f P ( E ) w h i c h i s g i v e n

b y t h e r e c i p r o c a l w i d t h o f e K ( t) . W h e n a s i s n o t t o o s m a l l

a n d T ~ > m i n { E c a , e -2 ~ , E c } t h e f u n c t i o n K ( t ) i s b a s i -

c a l ly d e t e r m i n e d b y K 1 ( t ). T h e n ( 2 1 ) a n d ( 2 2 ) i m p l y t h a t

t h e p r o b a b i l i t y o f i n el a s ti c t u n n e l i n g P ( E ) i s b r o a d e n e d

w i t h i n c r e a s in g T a n d decreasing a s i n a n i n c r e a s i n g e n -

e r g y r a n g e . I n d e e d i n t h i s c a s e t h e e n v i r o n m e n t c a n a c -

t i v a t e t h e t u n n e l in g . T h e c o n s e q u e n c e s a r e p r o n o u n c e d

i n th e q u a s i - p a r t i c l e c u r r e n t . I n t h is r e g i m e t h e g a p r e g i o n

i s s t r o n g l y s m e a r e d a n d t h e s u b g a p c o n d u c t a n c e i s e n -

h a n c e d c o m p a r e d t o t h e G i a e v e r re s u lt a t th e s a m e t e m -

p e r a t u r e , t h e s t r o n g e r t h e s m a l l e r ~ . I n th e n o r m a l s t a t e

t h e t u n n e l i n g c u r r e n t i n c r e a s e s m o n o t o n o u s l y w i t h i n -

c r e a s i n g a s . I n s u p e r c o n d u c t i n g j u n c t i o n s w e f i n d a m o r e

c o m p l e x b e h a v i o u r : a t v o l t a g es b e l o w t h e g a p a n d n o t

t o o s m a l l ~ , I qp i n c r e a s e s w i t h d e c r e a s i n g a s , t h i s te n -

d e n c y b e i n g i n v e r t e d a t l a r g e r v o l t a g e s .



4 5 6

T h e u s u a l p i c t u r e o f C o u l o m b b l o c k a d e is re c o v e r e d

f o r v e r y s m a l l e ~ cG < 0 .02 ) . In t h i s case [12 ] K t ) ~ K 1 t )

2c~vEc

is g iv en b y 23 ) fo r T > ~ l n e S 1 a n d d oe s n o t d e-

p e n d o n ~ , , w h e r e a s i f t h e l a t t e r c o n d i t i o n i s r e v e r s e d

2

K ( t ) ~ K o ( t ) ~ - r ~ ~ ~ v E c l n c ~ f I t 2,

as g i ven i n [12 ] , and

t h e w i d t h o f P E ) d e c r e a s e s w i t h d e c r e a s i n g cG .

5 a c v o l t a g e d r i v e

A s a n a p p l i c a ti o n o f t h e f o r m a l i s m d e s c r ib e d a b o v e , w e

s t u d y n o w t h e e f f e c t o n o f a n a c v o l t a g e a t t h e j u n c t i o n

V ( t ) = Vo + 1/i

c o s f ~ t o n t h e t u n n e l c u r r e n t . T h i s p r o b -

l e m h a s b e e n a d d r e s s e d i n [ 2 1 ] , i n t h e l i m i t

E j / E c ~> 1

w h e r e B l o c h o s c i l l a t i o n s [ 1 ] c o u l d m a n i f e s t t h e m s e l v e s .

I n t h e l i m i t c o n s i d e r e d h e r e , w h e r e p e r t u r b a t i o n t h e o r y

in

E j / E c

c a n b e u s e d , w e d o n o t e x p e c t a n y t i m e c o r -

r e l a t io n p h e n o m e n o n s i n c e t h e t u n n e l i n g is s t o c h a st i c a n d

t h e t i m e b e t w e e n t w o s u c c e s s iv e t u n n e l i n g e v e n t s is m u c h

l a r g e r t h a n a n y o t h e r t i m e s c a le i n t h e p r o b l e m .

N e v e r t h e l e s s t h e a c d r iv e le a d s t o f e a t u r e s i n t h e I - V

c h a r a c t e r i s t ic s . F o r a n a c v o l t a g e d r i v e n s y s t e m t h e c o r -

r e s p o n d i n g te r m in 9 ) a n d 1 0 ) b e c o m e s

s i n l i d s n e V ( s ) l : ~ , ~ J k (n v )J k ,( n v)

• V o ( t - t ' ) + ~? ( k t - k t ') }

w h e r e v = e

V1/ f2

a n d J ~ a r e B e s s e l f u n c t i o n s . T h e r e -

s u l t i n g d c s u p e r c u r r e n t a n d q u a s i p a r t i c l e c u r r e n t a r e

J ~ 2 v ) I v V k f 2 2 4 )

k = - - o o

k D ] 2 5 )

[ q p : ~ J l~ V ) [q p g o - - ~ -

k = - - o o

e s 0 . 8 4 . . . . . . . ~

0.4 '

0 .2

i i i i

o cv

e

F i g . 6 . T h e s u p e r c u r r e n t a t T = 0 , e , = l . 2 a n d

f 2 ~ E c

f o r

i n c r e a s i n g a c v o l t a g e

eV1/Ec=O,

1 . 5 , 3 . 0 , 4 . 5 f r o m l e f t t o r i g h t ,

r e s p e c t i v e l y . I n t h e i n s e t t h e s a m e c u r v e f o r e , = 0 .0 1 6 , ~ ~ 0 .1

E c

a n d

eV~/Ec=O,

1 0 0 , 2 0 0 , 3 0 0

w h e r e I v a n d Iq p a r e t h e d c c h a r a c t e r i s t i c s o f 1 6 ) a n d

1 5 ). T h e r e s u l t i n g I - V c h a r a c t e r i s t i c s a r e s h o w n i n

F i g . 6 . T h e e x p r e s s i o n f o r th e q u a s i p a r t M e c u r r e n t i s a g a i n

a s t r a i g h t f o r w a r d e x t e n s i o n o f t h e e q u i v a l e n t e x p r e s s i o n

f o r t h e n o r m a l s t a t e [ 22 ]. I n f a c t t h e y h a v e t h e s a m e f o r m

a s th e r e s u l t d e r iv e d b y T i e n a n d G o r d o n [ 23 ] a l o n g t i m e

a g o f o r c l a s si c a l j u n c t i o n s w h e r e c h a r g i n g e f f e c t s p l a y n o

r o l e . T h e c h a r g i n g e f f e c t s o n l y m o d i f y t h e f u n c t i o n I qp [ V ]

as g i ven in 15 ) .

6 R e m a r k s a n d e x t e n s i o n s

T h e n u m e r i c a l c a l c u l a t i o n s p r e s e n t e d a b o v e h e r e h a v e

b e e n c a r r i e d o u t o n l y f o r a p u r e l y O h m i c s e r i e s i m p e -

d a n c e , Z c o ) = R , . H o w e v e r t h e f o r m u l a t i o n a p pl ie s f o r

a n y l i n e a r e l e c t r o - m a g n e t i c e n v i r o n m e n t . T h e i n p u t w e

n e e d a r e t h e c l a ss i c a l I - V c h a r a c t e r i s t i c o f th e j u n c t i o n

a n d t h e i m p e d a n c e o f th e e n v i r o n m e n t . O t h e r c i r c u it s o f

i n t e r es t a r e , fo r i n s t anc e , t he R C l i ne [24 ] , o r t he L C li ne

w h i c h c a n m o d e l a s u p e r c o n d u c t i n g l in e . M o s t o f t h e

r e s u l ts p r e s e n t e d a b o v e a p p l y a l s o i n t h e l a t t e r c a s e i f o n e

p u t s

O~ s=R q(C/L) 1 /2 .

F o r i n s t a n c e , f o r

E ~ E c

t h e

f u n c t i o n

P o ( E )

h a s t h e s a m e f o r m a s i n t h e O h m i c

c a s e . O n t h e o t h e r h a n d , f o r

E>>8cGEc/Tr

w e h a v e

P0 E) oc exp

( , - ~ ( E - - E c ) 2 ,

b u t t h i s l e a d s o n l y t o

m i n o r c h a n g e s i n t h e I - V c h a r a c te r i st ic s a s c o m p a r e d

t o t h e r e s u l t s s h o w n a b o v e .

A n i n t e r e s t i n g q u e s t i o n i s w h i c h e n v i r o n m e n t f r e -

q u e n c i e s p l a y a ro l e in d e t e r m i n i n g t h e I - V c u r v e a t a

v o l t a g e V . A t s m a l l t e m p e r a t u r e s , E q . 1 5 ), 1 6 ) a n d 1 7 ),

t o g e t h e r w i t h t h e p r o p e r t y P 0 E ) = 0 f o r E < 0 s ug g e s t

t h a t t h e i m p o r t a n t f r e q u e n c i e s a r e c o < e V . I n d e e d t h e

f u n c t i o n a l f o r m o f t h e c h a r a c t e r i s t i c s i s d e t e r m i n e d b y

t h e s e f r e q u e n c i e s , b u t t h e n o r m a l i z a t i o n c o n d i t i o n o n

P 0 E ) i s n e e d e d i n o r d e r t o d e t e r m i n e t h e m u l t i p l i c a t i v e

c o n s t a n t . S o t h e I - V c u r v e is o b t a in e d f r o m f r e q u e n c i e s

u p t o t h e v a l u e c o r r e s p o n d i n g t o t h e e n e r g y a t w h i c h

P 0 E ) s t a r t s t o d e c a y , t h i s h a p p e n s f o r s m a l l E i f e s i s

l a r g e b u t f o r

E ~ E c

i f ev is smal l . I n t he l a t t e r ca se a l l

t h e f re q u e n c ie s o f t h e e n v i r o n m e n t u p t o

E c

a r e i m p o r t a n t

a n d a n i m p e d a n c e Z c o ) l a r g e u p t o co

~ E c

i s n e e d e d f o r

t h e C o u l o m b b l o c k a d e t o b e e f f ec t iv e .

A n o t h e r q u e s t i o n i s r e l a t e d t o t h e s m a l l v o l t a g e s b e -

d V

h a v i o u r . T h e t h e o r y p r e d ic t s -- , o e f o r V ~ 0 a n d T ~ 0

d I

i n n o r m a l t u n n e l j u n c t i o n s , w h e r e a s e x p e r i m e n t a l l y a f l a t-

t e n i n g t o w a r d s a n a p p a r e n t l y ) f i n it e v a l u e o f t h e d y -

n a m i c r e si s ta n c e is o b s e rv e d o n a p p r o a c h i n g T = 0 . T h i s

c o u l d b e a c c o u n t e d f o r b y t h e t h e o r y i f

P o ( E )

w o u l d

c o n t a i n a n e l a s t i c p e a k c o m p o n e n t , w h i c h c a n b e p r o -

d u c e d o n l y i f t h e v e r y l o w f re q u e n c y b e h a v i o u r o f Z ~ )

i s n o n - d i s s i p a t i v e . A t f i n i t e b u t l o w t e m p e r a t u r e s t h e d y -

nami c r es i s t ance i s [9 ]

2 F 3 / 2 + 1 /4 ~ s)

( 4 c ~ E c ' ~ 1 / 2~

R = R t

i r , / 2 F 1 + 1/4c~s) \ ~ /

w h i c h i s m u c h s m a l l e r th a n t h e v a l u e o b s e r v e d b y C l e l a n d

et al . [4].



A n e x t e n s i o n o f t h e t h e o r y t o l a r g e r tu n n e l i n g c o n -

d u c t a n c e s c a n b e a c h i e v e d b y t r e a t i n g t h e p a r t d e s c r i b in g

t h e t u n n e l i n g i n t h e a c t i o n ( 2 ) i n t h e h a r m o n i c a p p r o x i -

m a t i o n [ 1 6 ] , r a t h e r t h a n n e g l e c t i n g i t , a s d o n e i n t h e

e x p r e s s i o n ( 6 ) . T h i s a m o u n t s t o t h e s u b s t i t u t i o n

c q ~ a s + e t i n al l t h e f o r m u l a s a f t e r ( 7 ), b u t n o f u r t h e r

q u a l i t a t i v e c h a n g e s . N o t i c e h o w e v e r t h a t t h e h a r m o n i c

a p p r o x i m a t i o n i s c o r r e c t o n l y f o r l ar g e % + e t [ 2 5] .

I n t h e e x p e r i m e n t o f [ 5] t h e s u p e r c u r r e n t i s f o u n d i n

g o o d a g r e e m e n t w i t h t h e t h e o r y [ 1 2 ] a t v o l t a g e s o f o r d e r

V ~ e / C b u t i t s h o w s d e v i a t i o n s f o r s m a l l v o l t a g e s a n d

f o r e / C < V < 2 A . A t s m a l l v o l t a g e s w e e x p e c t s o m e

c o r r e c t i o n s d u e t o t h e f a c t t h a t q ~, c a n f e e l t h e e f f e c t o f

t h e w e a k w a s h b o a r d p o t e n t i a l t h a t i s n e g l e c t e d i n t h e

p e r t u r b a t i v e a p p r o a c h . A t l a rg e v o lt a g e s t h e e x p e r i m e n t a l

I - V c o n t i n u e s t o g r o w . I n c o n t r a s t t h e t h e o r y s u g g es t s

t h a t I s h a s t o f u l f i l s o m e n o r m a l i z a t i o n c o n d i t i o n ( s ee

( 1 6 )) . A c o n t r i b u t i o n t o t h e c u r r e n t a r i se s f r o m t h e s u b g a p

q u a s i p a r t i c le c u r r e n t . T h i s c a n b e c a l c u l a te d b y ( 1 5 ) o n c e

t h e s u b g a p c u r r e n t f o r a p e r f e c t l y v o l t a g e b i a s e d j u n c t i o n

i s k n o w n a n d c a n b e c o m e r e l e v a n t d u e t o t h e p r e s e n c e

o f th e e n v i r o n m e n t , a s r e m a r k e d i n S e c t. 4 . A n o t h e r c o n -

t r i b u t i o n c o m e s f r o m t h e q u a s i p a r t i c l e - p a i r i n t e r f e r e n c e

t e r m [ 1 4 , 1 8 , 2 6 ] ( t h e s o c a l l e d ' c o s i n e ' t e r m ) , w h i c h p r o -

d u c e s l o g a r i t h m i c d i v e r g e n c i es o f t h e c u r r e n t w h e n

V > _ 2 A .

U p o n d e c r e a s i n g e s o r i n c r e a s in g E j t h e t u n n e l i n g b e -

c o m e s c o r r e la t e d a n d , m o r e o v e r , c o h e r e n t. W h e n

a l /2 < E j / E c t h i s g i v e s r i s e t o B l o c h o s c i l l a t i o n s [ 1 w i t h

s

t h e t y p i c a l n o s e - s h a p e d c h a r a c t e ri s t ic s . T h e s p a c e o f p a -

r a m e t e r s a t T = 0 i s s h o w n i n F i g . 7 . T h e l i m i t t r e a t e d i n

t h i s a r t i c l e c o r r e s p o n d s t o t h e s h a d e d r e g i o n , w h e r e t h e

t u n n e l i n g i s s t o c h a s t i c . U p o n i n c r e a s i n g E j a n d / o r d e -

c r e a s i n g e s t h e c o h e r e n t r e g i o n i s m e t . I n t h e l a s t r e g i m e

t h e e n v i r o n m e n t p r o m o t e s Z e n e r t r a n s i t i o n s [ 2 7 ] i f t h e

c u r r e n t t h r o u g h t h e j u n c t i o n i s l a rg e . T h i s p r o b l e m h a s

b e e n t r e a t e d i n [ 2 8 ] i n t h e l i m i t c~ / a < E j / E c ~ 1 in a

c u r r e n t b i a s e d r e s is t iv e l y s h u n t e d j u n c t i o n , w h e n i n c r ea s -

i n g c~ s, Z e n e r t u n n e l i n g i n c r e a s e s t h e v o l t a g e d r o p a t t h e

j u n c t i o n . T h e c u r r e n t t h r o u g h t h e s h u n t t h e n i n c r e a s e s ,

b a l a n c e d b y a d e c r e a s e o f t h e c u r r e n t t h r o u g h t h e

j u n c t i o n . I n F i g . 8 w e c o m p a r e t h e I - V c u r v e s f o r

E / E c = O . 1 a n d d e c r e a s i n g a s . T h e y e v o l v e f r o m t h e

a s y m m e t r i c p e a k s h a p e i n t h e s t o c h a s t ic r e g i m e ( d a s h e d

l i n e ) t o t h e t i l t e d p e a k b e h a v i o u r a n d t h e n t h e ' B l o c h

n o s e ' o f th e c o h e r e n t r e g i m e . V i r t u a l p h o t o n e x c h a n g e ,

E j C o h e r e n t

- - t u n n e l i n g

EC /

Stochas tic

......... : ii : ~ m g : ....

t s

Fig. 7. The space o f parameters a s and E j / E c for a superconducting

junction is sho wn . Coherent tunneling occurs for Ey/Ec>oC~/2

whereas for larger ct, an d/ or smaller E j / E c the tunneling becomes

stochastic (shaded region)

457

IsRsC

e

1 . 5

0.5

;

/ / /

05 1 15

c v

% -

Fig. 8. The I - V characteristics in the stochastic and coherent re-

gime are compared. Here E j / E c = 0.1, and c%= 0.1 (dashed line),

0.0 l 5.0 10 4 2.5 10 -4 1.0 10-4

(solid lines) from left to right

respectively, they evolve from the asymmetric peak shape in the

stochastic regim e (dashed line) to the tilted peak b ehaviour and

then the 'Bloch nose' of the coherent regime (solid lines)

n e g l e c t e d i n [ 2 8 ] s h o u l d s m e a r t h e s h a r p f e a t u r e i n t h e

l a t t e r c u r v e s a t V ~ 0 . 8 7 e / C .

7 C o n c l u s i o n s

I n t h i s p a p e r w e s t u d i e d t h e e f f e c t o f t h e e l e c t r o m a g n e t i c

e n v i r o n m e n t o n t h e tu n n e l i n g o f q u a s i p a r t ic l e s a n d C o o -

p e r p a i r s i n s m a l l j u n c t i o n s . I f a h i g h i m p e d a n c e e n v i -

r o n m e n t i s o la t e s th e j u n c t i o n f r o m t h e s o u r c e , th e t u n -

n e l i n g i s r u l e d b y t h e d i f f e r e n c e o f th e loca l e n e r g y b e f o r e

a n d a f t e r t u n n e l in g , a n d t h e C o u l o m b b l o c k a d e s h o w s

u p . T h e k n o w l e d g e o f t h e c l as s ic a l I - V c u r v e f o r t h e

p e r f e c t l y v o l t a g e b i a se d e l e m e n t a n d o f t h e i m p e d a n c e o f

t h e c i rc u i t a l l o w s u s t o d e t e r m i n e t h e I - V c h a r a c t e r i st i c

i n a n e l e c t r o m a g n e t i c e n v i r o n m e n t . S i n g l e e l e c t r o n p h e -

n o m e n a a r e a p p a r e n t e v e n a t f i n i t e t e m p e r a t u r e . I n t h i s

r e g i m e t h e e n v i r o n m e n t c a n a c t i v a t e t h e t u n n e l i n g : t h i s

i s e v i d e n t f o r t h e s u b g a p q u a s i p a r t ic l e c u r r e n t w h i c h , d u e

t o t h i s e f f e c t , t u r n s o u t t o b e e n h a n c e d w i t h r e s p e c t t o

t h e c a s e o f p u r e l y v o l ta g e b i a s e d e l e m e n t f o r t e m p e r a t u r e s

a n d e x t e r n a l im p e d a n c e s o f e x p e r i m e n t a l i n t er e s t. T h e

t h e o r y c a n a l s o b e a p p l i e d t o a j u n c t i o n u n d e r r f i rr a -

d i a t i o n . F o r i n c r e a s i n g t u n n e l i n g c o u p l i n g s t r e n g t h a n d /

o r i n c r e a s in g i m p e d a n c e o f th e e n v i r o n m e n t t h e s t o ch a s -

t i c t u n n e l i n g b e c o m e s c o r r e l a t e d a n d f o r s u p e r c o n d u c t i n g

j u n c t i o n s e v e n c o h e r e n t . I n t h e l a t t e r c a s e t h e t i l t e d p e a k

s h a p e d I - V c h a r a c t e ri s t ic s s h o u l d p r o g r e s s i v e ly tu r n i n

t h e p e c u l i a r B l o c h ' n o s e ' b e h a v i o u r .

We w ould lik e to acknowledge stimulating discussions with D.A.

Averin, R. Fazio, L.J. G eerligs, U. G eigenmfiller~ G. G iaquinta,

D.B. Haviland, L.S. Kuzman, G.L. Ingold, K.K. Likharev, Yu.V.

Nazarov, A .A. Odintsov, A. Tagliacozzo, and A.D. Z aikin.

R e f e r e n c e s

1. Averin, D.V ., Likharev, K .K.: In: Mesoscopic phenomena in

solids. Altshuler, B .L, Lee, P.A., Webb, R .A. (eds.), Chap. 6.

Amsterdam: Elsevier 1991

2. Sch6n, G., Zaikin, A.D.: Phys. Rep. 198, 237 (1990)



458

3. Ku zm in, L.S., Delsing, P. , Claeson, T. , Likharev , K.K .: Phys.

Rev. Lett . 62, 2539 1989); Geerligs, L.J. , Andereg g, V.F. , van

der Jeugd, C.A. , R om ijn, J. , Mooij , J .E. : Eu roph ys. Lett . 10,

79 1989)

4. Martinis, J .M., Ka utz, R .L. : Phys. Rev. Lett . 63, 1507 1989);

Cleland, A .N. , Schm idt, J .M ., Clarke, J. : Phys. Rev. L ett . 64,

1565 1990)

5. Ku zm in, L.S. , Na zar ov, Yu.V. , Hav iland, D .B. , Delsing, P. ,

Cleason, T. : Prepr in t 1991; Kuz min, L.S., Ha vi land, D .B. : Pre-

print 1991, see also Havilan d, D.B. , K uzm in, L.S. , Delsing, P. ,

Likharev, K.K . , Claeson, T. : Z . Phys . B - C onden sed M at ter

85, 339 1991)

6. Geerligs, L.J., A ndereg g, V.F. , Ro mijn , J. , Mooij , J .E. : Phys.

Rev. Lett . 65, 377 1990)

7. M assen v an den Brink, A . , Sch6n, G . , Geerligs. L .J. : Phys. Rev.

Let t. to be publ i shed) ; B obber t , P . , Odintsov, A.A. : Europhy s .

Let t. submit ted for publ ica t ion) ; see a lso Ma assen van den

Brink, A. , Odintsov, A.A. , Bobbert , P.A. , Sch6n, G. : Z. Phys.

B - Condensed M at ter 85 , 459 1991)

8 . Naz arov, Yu .V. : Sov. Phys . JETP 68, 561 1989)

9 . Pan yuko v, S .V., Zaikin , A.D . : J . Low Tem p. Phys . 73 , 1

1988); Averin, D.V. , Odintsov, A.A.: Phys. Lett . A140, 251

1989)

10. Devo re t , M.H . , Es t~ve, D. , Graber t , H. , Ingold , G .L. , Pothier ,

H. , Urb ina, C. :Ph ys . Rev. Lett. 64, 1824 1990); Girvin, S.M.,

Glazm an, L. I . , Jonson, M . , Penn, D.R . , S t i les , M.D . :Phys . Rev.

Lett . 64, 3183 1990)

11. Fisher, M .P.A. , Zw erger, W .: Phys. Rev. B32, 6190 1985);

Zaikin, A.D. , Panyukov, S.V.: Phys. Lett . A120, 306 1987)

12. Aver in , D.V. , Na zarov , Yu.V., Odintsov, A .A. : Physica B1 65/

166, 945 1990)

13. Leggett , A.J . : Phys. Rev. B30, 1208 1984)

14. Eckern, U. , Sch6n, G. , Ambegaokar, V. : Phys. Rev. B30, 6419

1984)

15. Sch6n, G. : In : M acroscop ic quantum phenom ena. C lark , T .D. ,

Prance, H . , Prance, R .J. , Spiller, T.P. eds.) , p. 103. Sing apo re:

W orld Scienti fic 19 91; Aver in , D.V. , Sch6n, G . : NA TO A SI

Ser ies Quantum coherence in mesoscopic sys tems. Kramer , B.

ed .) , p . 531. New Yo rk: Plenu m Press 1991

16. Flensburg, K. , Jonso n, M .: Phys. Rev. B43, 7586 1991)

17. Falci , G. , Bubanja, V . , Sch6n, G. : E urop hys. Lett . to be pub -

lished)

18. see for instance T inkh am , M . : Intr odu ctio n to superconductiv-

i ty . Ne w Yo r k : M c Gr a w- Hi l l 1975

19. Minnh agen, P . : Ph ys . Let t. A56 , 327 1976)

20. Ingo ld, G.L. , Grabert , H . : Eu roph ys. Lett . 14, 371 1991)

21. Od intsov, A.A .: Sov. J. Low Temp. Phys. 14, 568 1988); Na-

zarov, Yu.V. , Odintsov, A .A. : Proceedings of the Co nference

SQ UID 91, Ber lin , June 1991 to be publ i shed)

22. Odintsov, A .A. : Fiz . Nizk. Temp. 15 , 466 1989) [Sov. J. Lo w

Temp. Phys. 15, 263 1989)]

23. Tien, P.K. , G ord on , J.P. : Phys. Rev. 129, 647 1963)

24. Naz arov, Yu .V. : JETP Let t. 49 , 126 1989)

25. Scalia, V. , Falci , G. , Fazio, R. , Gia quin ta, G . : Physica B16 5[

166, 975 1990); Phys. Rev. B subm itted for pub lication ); see

also Z. Phys. B - Co nden sed M atter 85, 427 1991)

26. Zaikin, A.D .: Prep rint 1991)

27. Mullen, K. , Ben-Jacob, E. , Schuss, Z. :Phys. Rev. Lett . 60, 1097

1988); GeigenmfiUer, U. , Sch6n, G. : Physica B152, 186 1988)

28. Bubanja, V. , M aasse n van den Brink, A. , Averin, D.V. , Sch6n,

G. : In : Gran ular nanoelec tronics . N AT O AS I Ser ies . Fer ry ,

D.K . ed .) , p . 567. New Y ork : Plenum Press 1991