TK 25.546

KFKI 15 /1967

/v * 1 1

i könyvtara *■ s

\^%ДГ0 INTÍl^У

GENERAL THEORY OF TUNNELING IN OXIDE DIODES

A. Zawadowski

HUNGARIAN ACADEMY O F SCIENCES CENTRAL RESEARCH INSTITUTE FOR PH YSICS

B U D A P E S T

A. Zawadowski

C e n t r a l R e s e a r c h I n s t i t u t e f o r P h y s i c s , B u d a p e s t, H ungary

The w e ll-k n o w n t h e o r y o f t u n n e l i n g i n o x id e d i o d e s i s th e t u n n e l i n g H a m ilto n ia n m eth o d , h u t t h i s c a n n o t d e s c r i b e p r o c e s s e s h a p p e n in g i n th e o x id e la y e r .S o m e new e x p e r im e n ts n e c e s s i t a t e th e t r e a t m e n t o f t h e e l e c t r o n s i n th e b a r r i e r a s w e l l . The a u t h o r h a s e l a b o r a t e d a m eth o d u s i n g G re e n ’ s f u n c t i o n s t o d e s c r i b e th e w hole phenom enon i n an i t e r a t i v e p r o c e d u r e . The s t a r t i n g p o i n t i s t h e t r e a t m e n t o f two o t h e r p ro b lem s w h ere th e m e t a l on th e l e f t o r r i g h t s i d e o f th e b a r r i e r i s r e p l a c e d by an i n s u l a t o r . The c u r r e n t d e n s i t y i n th e b a r r i e r h a s b e e n d e r iv e d f o r norm al an d s u p e r c o n d u c t in g j u n c t i o n s . The phenom enon i n a m a g n e tic f i e l d h a s b een t r e a t e d u s i n g th e m ic r o s c o p ic t h e o r y , a v o i d in g p h e n o m e n o lo g ic a l c o n s i d e r a t i o n s . The a p p l i c a b i l i t y o f th e t u n n e l i n g H a m ilto n ia n has b e e n i n v e s t i g a t e d ; b y i t s u s e th e t o t a l c u r r e n t may be c a l c u l a t e d . T h is m ethod h a s p r o v e d to b e v e r y s u i t a b l e f o r t h e p r o b ­ lem o f t h e anom alo us t u n n e l i n g b e tw e e n two n o rm a l m e ta ls w ith p a r a m a g n e tic i m p u r i t i e s i n th e b a r r i e r .

I . INTRODUCTION

I n r e c e n t y e a r s , th e p ro b le m o f t u n n e l i n g b e tw e e n two n o rm a l o r s u p e r c o n d u c t in g m e ta ls h a s b een i n v e s t i g a t e d t h o r o u g h l y i n num erous e x p e r i ­ m e n ta l and t h e o r e t i c a l w o rk s. The t h e o r y o f t u n n e l i n g th r o u g h a b a r r i e r was f i r s t i n v e s t i g a t e d b y B a r d e e n .^ The g e n e r a l f o r m a lis m o f th e p ro b lem h a s b e e n g iv e n b y Cohen, F a l i c o v , a n d P h i l l i p s , p who p ro p o s e d th e t u n n e l i n g H a m ilto n ia n . T h is m eth o d h a s p r o v e d t o be v e r y s u c c e s s f u l i n th e i n t e r p r e t a ­ t i o n o f e x p e r i m e n t a l r e s u l t s '.

I n th e t u n n e l i n g - H a m i l t o n i a n m ethod th e b a r r i e r i s r e p l a c e d b y a m a th e m a tic a l s u r f a c e and th e H a m ilto n ia n d e s c r i b e s p r o c e s s e s i n w h ic h an e l e c t r o n c r o s s e s th e b a r r i e r . T h is m ethod i s a r a t h e r p h e n o m e n o lo g ic a l one an d f a i l s to i n v e s t i g a t e th e t u n n e l i n g p r o c e s s e s t h e m s e lv e s . The d i f f i c u l t y i n th e e l a b o r a t i o n o f a new t h e o r y d e s c r i b i n g t h e e l e c t r o n s i n th e b a r r i e r

2

a s w e l l , com es fro m th e c h o i c e o f a s e t o f wave f u n c t i o n s t h a t i s c o m p le te a n d o r t h o g o n a l . T h is p ro b le m h a s b e e n s t u d i e d v e r y c a r e f u l l y h y P ra n g e , 3 and th e a p p l i c a b i l i t y o f t h e tu n n e l i n g - H a m i l t o n i a n m ethod h a s b e e n p r o v e d i n t h e f i r s t - o r d e r a p p r o x im a tio n . A q u i t e d i f f e r e n t a p p r o a c h h a s b e e n s u g ­ g e s t e d b y d e G e n n e s ,^ who h a s d e r iv e d a g e n e r a l i z a t i o n o f t h e G in s b u rg — L an d au e q u a t i o n f o r th e t u n n e l i n g p r o c e s s e s . R e c e n t l y , J o s e p h s o n p r o p o s e d Ц a v e r y s u g g e s t i v e m ethod u s i n g G re e n ’ s f u n c t i o n s , b u t i t seem s t o u s t h a t th e a c t u a l a p p l i c a t i o n o f t h i s m ethod i s n o t s i m p l e .

N e v e r t h e l e s s , a fe w e x p e r im e n ts h av e t u r n e d u p i n w h ic h th e r e g i o n o f th e b a r r i e r p l a y s a v e r y i m p o r ta n t r o l e , f o r e x a m p le , t h e g e o m e t r i c a l r e s ­ o n a n c e and b o u n d a r y e f f e c t i n a s u p e r c o n d u c t in g t u n n e l j u n c t i o n m e a su re d b y Tom asch^ an d th e e l e c t r o n s c a t t e r i n g o n p a r a m a g n e tic i m p u r i t i e s i n th e b a r -

7 8

r i e f i n v e s t i g a t e d e x p e r i m e n t a l l y by W y a tt and b y R o w ell a n d S h en . I n a d d i ­ t i o n , t h e p r o x i m i t y e f f e c t h a s a g r e a t im p o r ta n c e i n t u n n e l i n g .

A t h e o r y o f t u n n e l i n g b e tw e e n s u p e r c o n d u c t in g o r n o rm a l m e ta ls a c r o s s a n i n s u l a t i n g l a y e r i s p r e s e n t e d h e r e w h ich d e s c r i b e s th e phenom enon i n th e b a r r i e r a s w e l l . G r e e n ’ s f u n c t i o n s a r e u s e d to a v o id t h e p ro b le m o f . th e c o m p le te n e s s and o r t h o g o n a l i t y o f th e wave f u n c t i o n s a s f a r a s p o s s i b l e . The s t a r t i n g p o i n t i s th e t r e a t m e n t o f two d i f f e r e n t p ro b le m s w here th e m e ta l on th e l e f t ( o r lig h t- ) s i d e o f th e b a r r i e r i s r e p l a c e d b y an i n s u l a t o r . I n t h e s e p r o b le m s , r e f e r r e d t o a s l e f t and r i g h t p r o b le m s , t h e m ain p a r t o f th e b o u n d a r y e f f e c t s h a s b e e n ta k e n i n t o a c c o u n t . T h is m e th o d may be a p ­ p l i e d t o t h e c a l c u l a t i o n o f 't h e c u r r e n t d e n s i t y i n th e b a r r i e r , f o r i t

d e s c r i b e s t h e e l e c t r o n s i n t h e b a r r i e r a s w e l l . W ith o t h e r m e th o d s , o n ly t h e t o t a l c u r r e n t c a n be c a l c u l a t e d . T h ro u g h o u t th e u s e o f th e c u r r e n t d e n s i t y , t h e e f f e c t i n a m a g n e tic f i e l d may b e d e s c r i b e d i n a n a p p r o p r i a t e w ay.

The G re e n ’ s f u n c t i o n s o f th e o r i g i n a l p ro b le m a r e d e te r m in e d b y th e G re e n ’ s f u n c t i o n s o f t h e s e two l e f t and r i g h t p ro b le m s i n an i t e r a t i v e p r o c e d u r e ( S e c s . 2 and 3 ) . ( i n S e c . 4)w e g iv e th e c a l c u l a t i o n o f th e c u r r e n t d e n s i t y i n t h e b a r r i e r . T h ese r e s u l t s a r e a p p l i e d t o th e J o s e p h s o n c u r r e n t ( S e c . 5 ) , a n d t o th e l o n g - r a n g e o r d e r i n th e J o s e p h s o n j u n c t i o n i n a m a g n e tic f i e l d ( S e c . б ) . We d i s c u s s t h e a p p l i c a b i l i t y o f t h e tu n n e l i n g - H a m i l t o n i a n m e th o d , an d we c o n c lu d e t h a t t h e t u n n e l i n g H a m ilto n ia n i s a p o w e r f u l m e th o d f o r th e c a l c u l a t i o n o f t h e t o t a l c u r r e n t S e c . 7 • F i n a l l y , t h e p o s s i b i l i t y o f h i g h e r o r d e r p r o c e s s e s i s d i s c u s s e d v e r y b r i e f l y ( S e c . 8 ) .

2 . THE MATHEMATICAL FORMULATION OF THE PROBLEM

We m u st d e s c r i b e a n i n t e r a c t i n g e l e c t r o n g a s w h ic h i s d i v i d e d b y a p o t e n t i a l b a r r i e r i n t o two p a r t s , c a l l e d th e l e f t ( a nd r i g h t ( r ) s i d e s . The h e i g h t o f th e p o t e n t i a l b a r r i e r i s g r e a t e r t h a n th e F e rm i e n e rg y f o r a n

i n s u l a t i n g o x id e l a y e r . We s h a l l a p p l y th e m e th o d o f th e therm odynam ic.

G r e e n ’ s f u n c t i o n s . The n o rm a l and an o m alo u s o n e - p a r t i c l e G r e e n ’ s f u n c t i o n s i n t r o d u c e d b y G o rk o v ' a r e9

Сй ('л, *' )= < r { V « ( x ) У а ( х ' ) } ) •

a n d

£ а ъ ,х ' ) - < 4 f « ( x ) v ; ( x ' ) ) > •

w h ere у i s th e f i e l d o p e r a t o r o f th e e l e c t r o n f i e l d . The i n t e r a c t i o n o f th e e l e c t r o n s o r i m p u r i t i e s i s r e p r e s e n t e d by th e m a ss o p e r a t o r £ , w h ic h i s c a l c u l a t e d a c c o r d in g to th e s p e c i a l p ro b le m . We d e s c r i b e th e b a r r i e r a s a p o t e n t i a l V. F o r b r e v i t y , we i n t r o d u c e th e m a t r i x n o t a t i o n f o r th e G r e e n ’ s f u n c t i o n s G a n d F,

w h ere th e s u p e r s c r i p t T d e n o te s t h e ex ch an g e o f th e a r g u m e n ts , and t h e s p i n i n d i c e s w i l l n o t be w r i t t e n o u t .

The e q u a t io n o f m o tio n m ay be w r i t t e n a s

4 -

Z ) C - 1 ,

(3 )

w h ere Ga i s th e i n v e r s e o f th e n o n i n t e r a c t i n g - e l e c t r o n G re e n ’ s f u n c t i o n , w h ic h i s

*-/ ( 1 !rx0 + 7 ^ ^ ~v 0 \

I n t h i s f o r m u la Xo d e n o t e s th e tim e v a r i a b l e an d /и. t h e c h e m ic a l p o t e n ­ t i a l . The d e f i n i t i o n s o f th e i n v e r s e s a r e

G - ' G ~ i C 5 a )

an d

Л Л 4 ^Л

G G ' 1 = / / .

( 5 Ъ )

Л

w here / i s th e u n i t o p e r a t o r . The se c o n d i d e n t i t y may b e p r o v e n by p a r t i a l i n t e g r a t i o n s .

The c r u c i a l p o i n t o f o u r a p p r o a c h i s t h e r e d u c t i o n o f th e s o l u t i o n o f th e a b o v e -m e n tio n e d o r i g i n a l p ro b le m ( o . p ) w i t h th e p o t e n t i a l b a r r i e r

- 4 - 1

s e p a r a t i n g th e l e f t a n d r i g h t s i d e s t o th e s o l u t i o n s o f tw o o t h e r p r o b le m s . I n th e new p r o b le m s , t h e e l e c t r o n g a s i s l o c a l i z e d to t h e l e f t o r t h e r i g h t s i d e o f t h e b a r r i e r , i n t r o d u c i n g a p p r o p r i a t e p o t e n t i a l w e l l s F i g . 1 . These p ro b le m s a r e c a l l e d " l e f t and r i g h t p ro b le m s " ( l . p . and r . p . ) . The p o te n ­ t i a l s i n t r o d u c e d a r e c h o s e n i n s u c h a way t h a t th e p o t e n t i a l o f th e l . p . ( r . p . ) i s e q u a l to t h e p o t e n t i a l o f th e o . p . on th e l e f t ( r i g h t ) s i d e and i n s i d e o f th e b a r r i e r . The c o r r e s p o n d in g m a ss o p e r a t o r s and £ r т а У be c h o s e n i n a s i m i l a r w ay. T h ese d e f i n i t i o n s may be f o r m u l a t e d m a th e m a tic a l l y i f we i n t r o d u c e tw o s m o o th e d - o u t s t e p f u n c t i o n s a n d hn c o r r e s p o n d i n g to th e l e f t and r i g h t s i d e s , w h ich v a r y o n l y i n s i d e th e b a r r i e r [ F ig . I d ] , A ssum ing t h e i d e n t i t y

we c a n w r i t e10

ht t h r =■ f

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C 6)

and

Z - h c L e ♦ ! * ; £ • „ - Г „ л „ + Г , л ,

( ? )

These e q u a t i o n s a r e in d e p e n d e n t o f t h e s p e c i a l c h o ic e o f t h e s t e p f u n c t i o n s , and we s h a l l h av e t o show t h a t a l l p h y s i c a l r e s u l t s a r e in d e p e n ­ d e n t o f t h e i r c h o i c e .

L e t u s i n t r o d u c e th e G r e e n ’ s f u n c t i o n s and t h e i r i n v e r s e s f o r th e l . p . an d r . p . by th e d e f i n i t i o n s

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1 + + 'S T V«

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ő x 0 2 m +

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w here <x.= t , r and

Л у Л Л

Go~ Gok = 1 ( 9 a )

F u r th e r m o r e ,

A f A_, л

^ = G0<x ~ ( ю )

an d л л /\ л a , a

g; 1 / , • $ , £ ■ = /

( 9 b ) I t i s e a s y t o s e e fro m E q s . ( 4 ) , ( б ) , ( 7 ) , ( 9 a ) and ( 9b ) t h a t t h e f o l l o w i n g i d e n t i t i e s . a r e e x a c t :

and

G ~' -- Gn h. + G ' 1 h r

о ос t Or- r ^{( “ )}

л л л .

G - G~ he + G~ hr (

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I n s e r t i n g ( 12) i n t o ( 5 a ) a n d ( 5b ) and m u l t i p l y i n g b y Gr and Gt fro m t h e l e f t an d r i g h t , r e s p e c t i v e l y , and u s i n g (9 b )» we g e t th e f o l l o w i n g i d e n t i ­ t i e s :

у Л Л / ч л л

( h r + Gr Gt ht ) G =Gr ( l 3 a )

А л ^Л л

G ( h n + h { G{ Gr ) = Gr , ( l 3 b )

! $ "N 1 ( 1 3 c)

r <—» C cn ( / J a )

( 1 3 d ) A dding t h e s e f o u r e q u a t io n s t o g e t h e r , we g e t th e f u n d a m e n ta l e q u a ­ t i o n o f o u r a p p r o a c h , w h ere th e G re e n ’ s f u n c t i o n o f o . p . i s e x p r e s s e d b y th e G r e e n ’ s f u n c t i o n s o f th e l . p . and r . p . :

G = G r+Ge (GnG i 'he + r ~ L) G +ő(he G/' Gr + r-

_{- / ) / ( 1 4 )}

Our p ro g ram i s t o g iv e an a p p ro x im a te s o l u t i o n o f t h i s e q u a t i o n , s u p p o s in g t h a t th e t r a n s i t i o n r a t e o f e l e c t r o n s th r o u g h th e b a r r i e r i s v e r y s m a l l .

3 . THE ITERATIVE SOLUTION OF THE GREEN’ S - FUNCTION EQUATION

We t a k e th e F o u r i e r tr a n s f o r m o f 14 w i t h r e s p e c t t o tim e :

G ( E ) ~ c r ( E ) ' C ' ( n - H ( G r (E)C'-m /,'r r ~ o ä ( c b f t O ( h A ‘( E K ( £ ) N

t o make th e d i s c u s s i o n o f t h e p h y s i c a l b a s e o f o u r a p p r o x im a tio n e a s i e r . 11 E q u a tio n (14a) show s q u i t e d i f f e r e n t c h a r a c t e r i s t i c s a t s m a ll an d l a r g e v a l u e s o f t h e e n e rg y v a r i a b l e s . A t s m a ll v a l u e s o f th e e n e r g y , th e G re e n ’ s f u n c t i o n s G( and Gr a r e s t r o n g l y l o c a l i z e d to th e l e f t o r to t h e

- 6 -

r i g h t s i d e o f th e b a r r i e r . O nly t h e t a i l s o f th e G re e n ’ s f u n c t i o n s p e n e t r a t e i n t o th e b a r r i e r . I n t h i s c a s e t h e e x p r e s s i o n i n th e b r a c k e t i s v e r y s m a l l, f o r i t c o n t a i n s p r o d u c t s l i k e Gr G'c h c Л A w h ic h a r e p r o p o r t i o n a l to th e r a t e o f p e n e t r a t i o n i n t o t h e b a r r i e r . T h e r e f o r e Gc * Gr i s th e z e r o - o r d e r a p p r o x im a tio n to t h e G re e n ’ s f u n c t i o n n e a r th e Ferm i e n e r g y .

At h ig h v a l u e s o f th e e n e r g y v a r i a b l e , th e e f f e c t o f t h e p o t e n t i a l s Vc and \/r may be n e g l e c t e d . I n t h i s c a s e b o t h o f th e tw o te rm s 1 з th e

b r a c k e t o f E q. ( l 4 a ) a r e a p p r o x im a te ly t h e s o l u t i o n o f E q . ( l 4 a ) i t s e l f . We a r e i n t e r e s t e d i n t h e s o l u t i o n o f t h i s e q u a t i o n o n ly i n t h e f i r s t c a s e and i t seems r e a s o n a b l e t o s o lv e th e e q u a t i o n b y i t e r a t i o n s t a r t i n g w i t h th e z e r o - o r d e r a p p r o x im a ti o n Gg + Gr

To g e t r i d o f th e p h y s i c a l l y u n i n t e r e s t i n g p a r t o f th e G r e e n ’ s f u n c t i o n s w ith l a r g e v a l u e s o f t h e e n e r g y v a r i a b l e , we a p p l y a c u t o f f i n th e s p e c t r a l f u n c t i o n s o f th e G re e n ’ s f u n c t i o n s a t some e n e r g y b e tw e e n t h e to p o f th e p o t e n t i a l b a r r i e r and th e F e rm i e n e r g y F i g . 2 . The p rim e on t h e G reen ’ s f u n c t i o n w i l l d e n o te t h a t th e c u t o f f p r o c e d u r e i s a p p l i e d . I t i s worth m e n tio n ­ in g t h a t E q s . f 9 b ) a r e t o be c o r r e c t e d f o r t h e t r u n c a t e d G re e n ’ s f u n c t i o n s . The c o r r e s p o n d in g new e q u a t io n s a r e

f d tx-Q;'(x,xVC‘( x “x ’) = ^ ( x , x 1 = J(X„-х:,)&, ( \ X') # ^{I ,} C1?)

w here th e a p p l i c a t i o n o f th e c u t o f f r e s u lt s i n a sm eare d -o u t D ir a c d e l t a f u n c -

Л / \ Л

t i o n D . S in c e th e G re e n ’s f u n c t i o n s G1 a n d G a r e s t r o n g l y l o c a l i z e d to

Л у £ Г Л

th e l e f t o r r i g h t s i d e s , Dc i s v e r y s m a ll on th e r i g h t s i d e and D'r o n th e l e f t s i d e .

O ur fu n d a m e n ts ], e q u a t i o n ( lA a ) a f t e r th e a p p l i c a t i o n o f t h e c u t o f f i s

g

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;

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) +

c

;(

e

) -U (ő ;(E ) + r * ~ o ä ’(E )

+ GXE)(h'C;'(E)C'r (E) W ) / . (1 6 )

w here th e te r m s i n t h e b r a c k e t a r e s m a l l , b e c a u s e we h a v e s u p p o s e d t h a t th e p e n e t r a t i o n o f th e w ave f u n c t i o n i n t o th e b a r r i e r i s w e a k . By i t e r a t i n g t h i s e q u a t i o n , we g e t some t y p i c a l te r m s w h ich we w i l l now i n v e s t i g a t e .

I t w i l l be u s e f u l to t r a n s f o r m some te r m s , e . g . , i n th e f o l l o w i n g w ays,

л A - Л -Л / г Л / Л . / \ 1

a;c- he

g

;~

c

;(

l g

- \ L +

л ,^g

,

Л _ Л . л л л

- с

( 1 7 )

w h ere we h av e made u s e o f th e i d e n t i t y . ( 15)* an d [ ] - d e n o t e s th e commu­

t a t o r . The c o m m u ta to r i n ( 17) may b e c a l c u l a t e d u s i n g ( в ) a n d

( i o )

^:

[ = r r y - ' ( ^ h „ ) % + ( , 2 т У Ä h ^ - l y h J - , ( 1 8 )

w h ere th e d i r e c t i o n o f a n a rro w above a d i f f e r e n t i a l o p e r a t o r i n d i c a t e s th e o p e r a n d . The c y c l i c r u l e i s to be f o l l o w e d i n t h e a b s e n c e o f an a d j a c e n t o p e r a n d .

We s u p p o s e t h a t t h e co m m u tato r on th e r i g h t s i d e o f ( l 8 ) v a n is h e s »

(19)

t h e i d e n t i t i e s i n E q. ( 19) a r e f u l f i l l e d i f th e m ass o p e r a t o r s a r e l o c a l f u n c t i o n s i n t h a t r e g i o n o f sp ace w h ere th e c o r r e s p o n d in g s m o o th e d - o u t s t e p f u n c t i o n s v a r y , i . e . , i n s i d e th e b a r r i e r ( s e e R e f . IO ).

I n s e r t i n g ( l ö ) a n d ( 1 9 ) i n t o ( 17) , we g e t f i n a l l y

= O X ^ ' ( % hc)

7

. +(2r#1(ZhX)'G't X ; heD (

20

)

I t i s e a s y t o p r o v e th e f o l l o w i n g i d e n t i t y f o r a n o t h e r t y p i c a l te r m a p p e a r in g i n th e i t e r a t i o n o f ( 1 б ) :

g: g; h , g: = Ó ; ( X - ( A V( -- A ) - ( Л L e-- A E j ) h c G i

= 4, ht c ; -

g

; ( a ve - л vn) л, £ л

- c ' r ( & £ r b t ) h c c y

(

²¹

)

w h ere we have i n t r o d u c e d t h e new n o t a t i o n

AV* = V* - V and

a

£c( =

_{Е в -}

L, (

2 2

)

a n d th e f o l l o w i n g i d e n t i t y h a s b een a p p l i e d :

S -'-c t:' = (v,-.\/r )* ( c , - z } ( 2Я

w h ic h f o llo w s f r o m ( в ) a n d ( l O ) . I t i s w o r th m e n tio n in g t h a t A V ( i s d i f ­ f e r e n t from z e r o o n ly on th e r i g h t s i d e o f th e b a r r i e r . S i m i l a r r e s u l t s may b e o b ta in e d b y t h e e x ch a n g e o f th e i n d i c e s r an d L .

The r e s u l t i n t h e f i r s t - o r d e r i t e r a t i o n o f (1 б ) may b e w r i t t e n a s t h e аглп o f f o u r te r m s w i t h d i f f e r e n t p h y s i c a l i n t e r p r e t a t i o n s :

Л / \ Л Л A A

G '= g; + G'n + 6 G T + <fCp +6Gd . ⁽24)

8

We w i l l g i v e th e o r d e r o f th e p a r t i c u l a r te rm s i n p o w ers o f th e t u n n e l i n g r a t e t . The t u n n e l i n g r a t e i s t h e r e l a t i v e d e c r e a s e o f t h e wave f u n c t i o n s a t th e F erm i e n e rg y i n th e b a r r i e r s

t = e x p f - [ 2 m ( \ / - /u ) ] i d / ( 2 5 )

w here d d e n o te s t h e t h i c k n e s s o f th e b a r r i e r a n d V i s t h e e n e r g y o f th e to p o f th e b a r r i e r .

The p a r t i c u l a r te rm s o f th e G re e n ’ s f u n c t i o n a r e : />

( 1 ) The t u n n e l i n g te r m , d G j :

cfGT( x , x ’) =j j i G (' ( x , y ) ( 4 , X1) ( 26 )

~ G ; ( x , & ( y > * ’) } x C % ^ t ( y ) ) i d 4' y ,

w here h r i s e l i m i n a t e d by h^ u s i n g ( б ) . I n A ppendix A a n o t h e r fo rm o f t h i s te rm i s d e r i v e d t o e l i m i n a t e h( a s w e l l :

cfGT( x , x ' ) = J ^ 1 <Ус'(г,у ) ^ - 2 m C ’' ( y , x ) - r ~ t J d ef 4 t l dL/0> ( 2 7 )

w here th e i n t e g r a l i s ta k e n on an a r b i t r a r y s u r f a c e «5" l y i n g i n t h e b a r r i e r . The s u r f a c e e le m e n t v e c t o r d f y , i i s d i r e c t e d fro m l e f t to r i g h t , and i t may be shown t h a t t h e v a l u e o f t h i s i n t e g r a l i s in d e p e n d e n t o f t h e s p e c i a l c h o ic e o f th e s u r f a c e S to a good a p p r o x im a ti o n . N am ely, we c a n p r o v e t h a t t h a t p a r t o f cfGT w h ich i s d e p e n d e n t o n th e c h o ic e o f th e s t e p f u n c t i o n h i s s i m i l a r t o t h e ' t h i r d ty p e o f c o r r e c t i o n cfG£ [ s e e E q . ( 3 3 ) an d A ppendix A ], w h ich i s a lw a y s n e g l e c t e d .

I t i s e a s y t o s e e t h a t t h i s te rm c o r r e s p o n d s t o t h e o n e - p a r t i c l e t u n n e l i n g and t h e r e f o r e i s p r o p o r t i o n a l t o t h e t u n n e l i n g r a t e t , a n d t h a t i t s p a r t d e p e n d in g on th e c h o ic e o f h i s o f o r d e r t z . These c o r r e c t i o n g iv e th e c o u p lin g b e tw e e n th e G re e n ’ s f u n c t i o n s c o r r e s p o n d in g to t h e two d i f ­ f e r e n t s i d e s of th e b a r r i e r by th e c u r r e n t c o u p l in g d e r i v e d by B a r d e e n . 1- They may be i l l u s t r a t e d by th e d ia g ra m s i n F i g . 3»

( 2 ) The r e n o r m a l i z a t i o n te rm s <fGR due t o t h e p o t e n t i a l and mass o p e r a t o r c o r r e s p o n d in g t o th e o p p o s i t e s i d e s :

<SG, -р Ё ' Л щ - a vr ) h cc ; ? c ; h t ( ä v r - A vr ) £■

+ c ; ( a t t - /

¹

1 ) hr a ; + c; ht ( a t , - л £ r ) &'„} d ( r - " 1}

U sin g th e d e f i n i t i o n s o f AV( , A V r , a T-c a n d A 'L r , t h e f o l l o w i n g i d e n t i t i e s may be p r o v e n :

( 2 9)

^ h ^ á t ^ O ( 3 0 )

I n s e r t i n g ( 2 9 ) a n d ( 5 0 ) i n t o ( 2 8 ) a n d m aking u s e o f ( б ) we h a v e g o t t e n r i d o f th e s m o o th e d - o u t s t e p f u n c t i o n s , h c a n d h r :

'5Ge (x,x') = -f/É ; ( x .y ) ü M r( y , 4 ')G;(4 ' , x ' ) + ( r ~ 0 } d >y d % ’ (31)

H ere we have i n t r o d u c e d th e new n o t a t i o n

й ! % ( Ч ,ч' ) - М « (у) * (ч- Ч ' ) + а£ « ( у > Ч' У ( 5 2 ) A c e r t a i n p a r t o f o G Rл p r o v i d e s a c o n t r i b u t i o n to t h e p r o x i m i t y e f -

A 5

f e e t . cTGr i s p r o p o r t i o n a l to th e s q u a r e o f t h e t u n n e l i n g r a t e t , an d i s r e p r e s e n t e d b y th e d ia g ra m s i n P i g . 4 .

( 3 ) Term s c o r r e s p o n d in g to t h e n o n o r t h o g o n a l i t y :

tfCj, ( x , x ') - - 1 ( f t G ^ x , 4 ) h c ( 4 ) De (tf, x ’) d \ * f i t Cx, 4 ) h t ( y ) G j y , x ) d ^ +

+j~G'r (x, фЬс((j)Dn ( y, x')d*y +f-Dr (x, y ) h c Cy)G^(.y, x ’j d j

These te r m s a r e s l i g h t l y d e p e n d e n t on th e c h o ic e o f th e a m e a r e d - o u t s t e p f u n c t i o n s . A c a r e f u l a n a l y s i s o f th e s e te r m s shows t h a t t h i s c o r ­ r e c t i o n r e n o r m a l i z e s th e o n e - p a r t i c l e wave f u n c t i o n s i n s i d e t h e b a r r i e r and i n th e n e ig h b o rh o o d o f i t . These c o r r e c t i o n te r m s a r e v e r y s t r o n g l y o s c i l l a t i n g o u t s i d e th e b a r r i e r and q u i c k l y damp w i t h i n c r e a s i n g d i s t a n c e fro m th e b a r r i e r . They a r e th e e f f e c t o f b r e a k s i n th e o r t h o g o n a l i t y a n d com- p le te n e s s o f th e u s e d o n e - p a r t i c l e wave f u n c t i o n s , d i s c u s s e d b y P ra n g e .T.

B ecau se t h i s c o r r e c t i o n i s l i k e a r e n o r m a l i z a t i o n o f th e wave f u n c t i o n s , th e d i r e c t c o n t r i b u t i o n to c u r r e n t v i a ( 34-) i s z e r o , b u t i n h i g h e r - o r d e r a p p r o x i ­ m a tio n s i t m ig h t g i v e a c o r r e c t i o n t o th e c u r r e n t p r o p o r t i o n a l t o i 3 . We c o n j e c t u r e t h a t t h e c o r r e c t i o n te rm s a r e c o n c e rn e d w ith some m a th e m a tic a l p ro b le m s o f o u r a p p r o a c h an d n e v e r w i t h some r e a l p h y s i c a l p r o b le m s . We w i l l n e g l e c t them i n t h e f o l l o w i n g .

We may g e t h i g h e r - o r d e r a p p r o x im a tio n s t o th e G r e e n ’ s f u n c t i o n s i n t h i s way u s i n g t h e c o r r e c t i o n s o f t h e f i r s t an d se c o n d t y p e .

- 10 -

The s u r f a c e a n d th e p r o x i m i t y e f f e c t may be t a k e n i n t o a c c o u n t i n tw o s t e p s :

( 1 ) The G r e e n ’ s f u n c t i o n s h av e b e e n c a l c u l a t e d i n th e c a s e w here th e m e ta l o n th e o p p o s i t e s id e an d th e b a r r i e r a r e r e p l a c e d by a s i n g l e i n s u l a t o r . I n t h i s way, t h e d e c r e a s e o f th e g ap f u n c t i o n n e a r th e b a r r i e r a n d th e s u r f a c e e f f e c t s h a v e b e e n t a k e n i n t o a c c o u n t .

( 2 ) A c t u a l l y , t h e r e i s a n o t h e r m e t a l b e h in d t h e b a r r i e r an d t h i s may c a u s e a s l i g h t m o d i f i c a t i o n o f th e gap f u n c t i o n n e a r t h e b a r r i e r . T h is c a n be c a l c u l a t e d i n p e r t u r b a t i o n t h e o r y u s i n g th e d ia g ra m s i n F i g s . 3 and 4 .

4 . THE CURRENT DENSITY IN THE BARRIER

The c u r r e n t d e n s i t y i n a n a r b i t r a r y p o i n t X c a n be c a l c u l a t e d u s i n g th e G r e e n ’ s f u n c t i o n .

Jt ( x ) ■= Urn { e ( ( V x - 4S ) / 2 m ) t G ' ( x , * ) ) X0~ Х0 +0,%~Х,-

О )

The z e r o t h - o r d e r a p p r o x im a tio n to G w i l l n o t g iv e any c o n t r i b u ­ t i o n to t h e c u r r e n t d e n s i t y . U sin g th e f i r s t - o r d e r a p p r o x im a tio n t o t h e G re e n ’ s f u n c t i o n s , we g e t th e l e a d i n g te rm o f c u r r e n t d e n s i t y , w h ic h i s g i v ­ e n by d ia g ra m s i n F i g . 5» w here t h e lo w e r - c a s e j ( x ) r e p r e s e n t s t h e c u r r e n t o p e r a t o r . The c o r r e s p o n d in g m a th e m a tic a l e x p r e s s i o n i s

(35)

I n th e s t a t i s t i c a l m e c h a n ic s of n o n e q u i lib r i u m p r o c e s s e s , th e c u r r e n t d e n s i t y i s c a l c u l a t e d a s a r e s p o n s e t o a n e x t e r n a l f o r c é ; i n t h e p r e s e n t c a s e , i t i s c a l c u l a t e d a s a r e s p o n s e t o th e t u n n e l i n g c o u p l i n g !

j j x ) = f o 'V * ’, i ’J_ d 4o K i ( X , x ' ) Tt > ( Л’) , ( 3 6 ) w here T i s th e g e n e r a l sym bol o f th e t u n n e l i n g c o u p l in g . The c a u s a l k e r n e l Ис ( х , х ’) i s c a l c u l a t e d u s in g t h e c a u s a l G r e e n ’ s f u n c t i o n s , b u t h e r e we n e e d th e r e t a r d e d o n e . I n v e s t i g a t i o n o f th e a n a l y t i c a l p r o p e r t i e s o f th e k e r n e l s show s ~] P t h a t t h e r e t a r d e d k e r n e l may b e o b ta in e d b y s h i f t i n g th e p o l e s o f t h e F o u r i e r tr a n s f o r m o f Kc ' below t h e r e a l a x i s i n th e com plex e n e r g y p l a n e . I f th e o p e r a t o r ( C — R ) s ta n d s f o r t h i s o p e r a t i o n , t h e n th e e x p r e s s i o n o f th e c u r r e n t d e n s i t y i s f i n a l l y

- r — t j (

^{5 7}

)

The c u r r e n t d e n s i t y d e r iv e d h e r e s a t i s f i e s th e e q u a t i o n o f c o n t i ­ n u i t y i n th e b a r r i e r . T his c a n be shown by a c a l c u l a t i o n s i m i l a r t o th e one i n A p p en d ix A. We w i l l r e t u r n l a t e r t o d i s c u s s t h e c o n n e c tio n o f ( 3 7 ) and th e c u r r e n t f o r m u la d e r iv e d b y u s in g t h e t u n n e l i n g H a m ilto n ia n .13

A s i m i l a r fo rm u la c a n be o b t a i n e d i n th e c a s e o f an e x t e r n a l mag­

n e t i c f i e l d :

Ji M = (C R ) e f d % t J d x o

J J , (3 7 a)

i(e/c) A(x)J - [ VA~ i(e/c)A(x)])^ ^ fc(e^)A (x)]-lfyd(eAOA(x)JJ

- r ~ ' J w here A ( x ) s ta n d s f o r th e v e c t o r p o t e n t i a l .

5 . JOSEPHSON CURRENT

To c a l c u l a t e th e a c t u a l v a lu e o f th e c u r r e n t , we h av e t o i n s e r t th e m a t r i x form o f th e G re e n ’ s f u n c t i o n s ( 2 ) i n t o th e e x p r e s s i o n o f th e c u r ­ r e n t ( З7 )» Then u s i n g th e s y m b o lic n o t a t i o n T f o r th e c o u p lin g c o n s t a n t we o b t a i n th e f o l l o w i n g fo rm u la

' j,(x)^ejTG'TCdTr;T^'J-ejr— c ] - ( 3 8 )

I t i s w e l l known t h a t th e a n o m alo u s G re e n ’ s f u n c t i o n s a r e d e p e n ­ d e n t o n th e s p e c i a l c h o ic e o f p h a s e s an d th e a b s o l u t e v a lu e o f t h e tim e a r g u m e n ts . T h e r e f o r e we may w r i t e

£ ( x , x ) = e x p ( - 2ÍJUL*X0) <pc!

( x , x '

, x0 - * ; )

* е х р ( - ( 2 е / с ) 1 ^ )

and ( 39)

F ^ \ x \ x) = exp(+ 2iy* xc) <f>J’( x \ X ; x j - x 0)

x exp

[+(2

e / с ) i p « ] ,

w here //<* i s th e c h e m ic a l p o t e n t i a l and </>* i s t h e p h a s e o f t h e p a i r wave f u n c t i o n s on th e cx s id e o f t h e b a r r i e r . H ere (f><x (°< = i,r) a r e in d e p e n d e n t o f th e c h o ic e o f p h a s e s .

The c u r r e n t d e n s i t y may .be w r i t t e n a s t h e sum o f two t e r m s :

j ( x ) =jN ( x ) + j j ( x )

^{(4 0 )}

12

w here

j „ < x ) * T G ; T G ' t - ( r ~ 0 (* 1а)

i s th e o n e - p a r t i c l e c u r r e n t d e n s i t y and

J j ( x ) = e x p { 2 [ ( e / c } A < p +

A ^ x 0 ] J

T ф'г Тф{' (41b) i s th e c u r r e n t d e n s i t y due t o t h e p a i r t u n n e l i n g , f i r s t s u g g e s te d by Josephson."*"^ H ere we hav e u s e d th e n o t a t i o n s

4 “

^{- A ( = e V ,}

(чг)

w here V i s th e a p p l i e d v o l t a g e , and

A<p = <pe -ipr ^

The a c t u a l v a lu e o f t h e c u r r e n t d e n s i t y i s g i v e n i n A p p en d ix B.

At z e r o a p p l i e d v o l t a g e , th e e x p r e s s i o n f o r th e c u r r e n t d e n s i t y r e d u c e s to th e f o l l o w i n g :

JJ

f

Jj,o sfn[ 2 (e /c )

A

p]

(4 4 )

6 . JOSEPHSON EFFECT IN MAGNETIC FIELD

We w i l l a l s o v e ry b r i e f l y t r e a t t h e J o s e p h s o n e f f e c t i n th e p r e s e n ­ ce o f a n e x t e r n a l m a g n e tic f i e l d . T h is t r e a t m e n t i s b a s e d on th e com pensa­

t i o n o f t h e lo n g r a n g e p h ase m o d u la tio n o f th e p a i r wave f u n c t i o n f ’ by an a p p r o p r i a t e t r a n s f o r m a t i o n d i s c u s s e d by t h e a u t h o r i n t h e c a s e o f f l u x o i d q u a n t i z a t i o n . 15^ I n s i d e l a r g e s u p e r c o n d u c t o r s th e m a g n e tic f i e l d v a n i s h e s ! th e v e c t o r p o t e n t i a l A L may b e w r i t t e n a s a g r a d i e n t o f a f u n c t i o n y5« , i . e . ,

/ 4 « , i ( X ) * 4 <P«(X) + /it (cx = l , r ) (45) w here S A# v a n i s h e s e x c e p t a t th e s u r f a c e l a y e r s a n d a t th e n e ig h b o u rh o o d o f th e b a r r i e r w h ere th e m a g n e tic f i e l d a p p e a r s . I t i s u s e f u l t o a p p l y th e f o l l o w i n g t r a n s f o r m a t i o n :

G l ( x . x ' ) = £ « ( * , x) ex p f - i ( e / c ) [ ЧЪ(х)- % (X')]J >

F« \x , x ) = eXp(+ 21p« Xo) l + K x \ X; x 0’ - x0)

x e x p { - L ( e / c ) [ (X ) t tp« (y, ’) ] } >

Ъ ' ( х . х ' ) = e x p C - ^ i j L L ^ x J ф « ( х , x ) X0 - x j )

X

e x p { - 1(е/сУ[<рч(х) + %f ( x ) ] }

( 4 6 )

T h is t r a n s f o r m a t i o n h a s a f o r m s i m i l a r t o t h a t o f a gauge t r a n s ­ f o r m a t i o n ; t h e r e f o r e , u s i n g th e g au g e i n v a r i a n t s t r u c t u r e o f o u r a p p r o a c h , i t i s e a s y to s e e t h a t i n th e s y ste m o f e q u a t i o n s f o r t h e G r e e n ’ s f u n c t i o n s o f t h e p a r t i c u l a r p ro b le m s o n ly сГАы o c c u r s , and i n th e s y ste m o f e q u a ­ t i o n s o f th e o r i g i n a l p ro b le m <fA^ a n d A ф( Х) - yV (x)_<Pr OO . ^ I n s i d e t h e s u p e r c o n d u c t o r s a n d ф<х s a t i s f y f i e l d - f r e e e q u a t i o n s , and so th e y a r e e q u a l t o th e G re e n ’ s f u n c t i o n i n th e a b s e n c e o f an e x t e r n a l f i e l d a n d w i l l be d e n o te d b y G0cC ’ and <6ŰC< ’ l a t e r i n t h i s s e c t i o n . We i n t r o d u c e a n o t a t i o n s i m i l a r to ( 4 5 ) ,

Oc = G0>J t and фа = ф01 J + S£ } (47)

w h ere d'Gcx a n d d a r e th e d e v i a t i o n s fro m th e f i e l d - f r e e G re e n ’ s f u n c ­ t i o n s . These d e v i a t i o n s a r e in d u c e d b y th e v e c t o r p o t e n t i a l d A * a c c o r d i n g t o th e M e is s n e r e f f e c t , a n d p ro d u c e t h e c u r r e n t w h ich c a n c e l s th e m a g n e tic f i e l d i n th e s u p e r c o n d u c t o r and s u p p l i e s th e c u r r e n t i n th e b a r r i e r . c f A a h a s t o be d e te r m in e d i n a s e l f c o n s i s t e n t way a s d is c u s s e d b y F e r r e l a n d P r a n g e . 17' T h is v e c t o r p o t e n t i a l i s s m a ll and t h e r e f o r e c a n b e t r e a t e d i n p e r t u r b a t i o n t h e o r y .

We c a n c a l c u l a t e th e J o s e p h s o n c u r r e n t v e r y s im p ly i f we s u p p o s e 1 D

t h a t th e p h a s e d i f f e r e n c e A f i s s lo w ly v a r y i n g i n th e b a r r i e r . I n s e r t i n g ( 4 6 ) i n t o (41b) a n d r e p l a c i n g A<p(x‘) by A p C x ) , we h a v e t h e f o r m u la f o r t h e J o s e p h s o n c u r r e n t d e n s i t y :

j j ( - x ) = e x p ( 2 e i ( c ' , A jp ( x ) + Vx0) } х Т ф 0/Г' ТфоХ’ - ( r ~ t ) C^8 ) A t z e ro a p p l i e d v o l t a g e t h i s f o r m u lá becom es m ore sim p le a s ( B l l ) h a s b e e n s i m p l i f i e d t o (В 1 2 ):

J

^j

( X ) = Jj

( x ) Ы п [ ^ ( е / с ) А

pCx)]

, (

49

) w h ere some p a r t o f th e p h a s e s h i f t A <p i s d u e t o th e m a g n e tic f i e l d a t th e j u n c t i o n . I f A

<p

c h a n g e s b y (J c / e ) n w here n = + 1, + 2 . . . , th e J o se p h so n ., c u r r e n t d e n s i t y d o e s n o t a l t e r .

We w i l l t r e a t t h e c o n n e c tio n b e tw e e n t h e d i r e c t i o n o f th e J o s e p h s o n c u r r e n t and t h e m a g n e tic f i e l d e n c l o s e d by th e j u n c t i o n w h ic h h a s b e e n d i s - c u s s e d by A n d e rs o n . 7 I n F i g . 6 we h a v e i l l u s t r a t e d a j u n c t i o n and th e 19

p e n e t r a t i o n o f th e m a g n e tic f i e l d i n t o i t . The p e n e t r a t i o n d e p t h s a r e d e n o t ­ e d by

A

^{£ a n d}

Ar

, r e s p e c t i v e l y . The m a g n e tic f i e l d may b e fo u n d f ro m th e m a g n e tic f i e l d H /w h ic h i s d i r e c t e d a lo n g th e a x i s у / \

Ax ( x,

z ) =y^

H4 ( x,

z )

d z

( 5 0 )

14 -

The a p p r o p r i a t e t r a n s f o r m a t i o n s d i s c u s s e d b e f o r e a r e d e te r m in e d b y th e f u n c t i o n s

X / • * r ° °

'Pc ( x ) ~ d x d z . H ( x , z ) and tpr ( x ) = / d x d z H 4 ( x , z } > ( 5 l )

Jxo Jo J x o Jo 7

w h ere X 0 d e t e r m in e d th e b o u n d a ry c o n d i t i o n f o r th e r e l a t i v e p h a s e s . The c o r r e s p o n d in g ch an g e o f th e p h ase d i f f e r e n c e b e tw e e n th e p o i n t s and i s

Д<р(хг ) - A < p ( x , ) d x j d z H v ( x , z ) = А ф н ( х г , Х ' ) ( 52) w h ic h i s th e m a g n e tic f l u x e n c lo s e d b y th e d a s h e d l i n e i n F ig .- 6 . The

J o s e p h s o n c u r r e n t i s u n a l t e r e d i f t h e f l u x c h a n g e s by пфн 0 w here ' ‘ IQ

фи 0 =Яс/е i s th e f l u x quantum , a s h a s b e e n f o u n d by A n d e rs o n . 7 The c u r r e n t d e n s i t y (49) and th e p h a s e d i f f e r e n c e (52) d e t e r m in e th e t o t a l c u r r e n t a s a f u n c t i o n o f th e e n c l o s e d m a g n e tic f i e l d , w h ich i s s i m i l a r t o a F r a u n h o f f e r i n t e r f e r e n c e p a t t e r n f o r m u l a .20

I t m ig h t be e x p e c t e d i t h a t a s i m i l a r i n t e r f e r e n c e e f f e c t w o u ld o c c u r i n th e l o c a l c u r r e n t d e n s i t y a t a f i x e d p o i n t x . One o f t h e e l e c t r o n s o f a t u n n e l i n g p a i r c r o s s e s th e b a r r i e r a t p o i n t x , b u t th e o t h e r one c r o s s e s som ew here e l s e i n th e r e g i o n o f th e c o h e re n c e l e n g t h a ro u n d th e f i x e d p o i n t

x . The p h a s e d i f f e r e n c e f o r th e s e c o n d e l e c t r o n m ig h t s t r o n g l y v a r y a s a f u n c t i o n o f th e t u n n e l i n g p la c e i f t h e m a g n e tic f i e l d w ere s t r o n g e n o u g h .

Then th e i n t e g r a n d o f th e c u r r e n t d e n s i t y e x p r e s s i o n (3 7 a ) w ould o s c i l l a t e a s a f u n c t i o n o f эс’ . I n f a c t , t h i s e f f e c t c a n n o t be o b s e r v e d b e c a u s e th e r e q u i r e d m a g n e tic f i e l d would- be c o m p a ra b lé w i t h th e c r i t i c a l m a g n e tic f i e l d .

✓

7 . THE TUNNELING HAMILTONIAN METHOD

The t u n n e l i n g H a m ilto n ia n h a s b een p r o p o s e d by C ohen, F a l i c o v and P h i l l i p s ^ t o d e s c r i b e th e e l e c t r o n t r a n s i t i o n s th r o u g h t h e b a r r i e r i n a p h e­

n o m e n o lo g ic a l way. The H a m ilto n ia n c o n t a i n i n g t h e f i e l d o p e r a t o r s o f b o t h s i d e s o f th e b a r r i e r i s

Н т = 'И Т Х',Х;С,г<*Я,1**\.г + COnj.

д,л' '

The t r a n s i t i o n a m p litu d e T i n th e t ű n n e l i n g - H a m i l t o n i a n m e th o d h a s b e e n f i t ­ t e d to th e e l e c t r o n s c a t t e r i n g a m p lit u d e c o r r e s p o n d in g to t h e t r a n s i t i o n from one s id e o f th e b a r r i e r t o th e o t h e r . A c c o rd in g t o B a r d e e n 's i n v e s t i g a t i o n s w h ic h a r e i n a g re e m e n t w i t h o u r r e s u l t s , PI t h e y a r e

fK x ; ! , r ( 5 J )

w h ere th e ' s a r e t h e o n e - e l e c t r o n wave f u n c t i o n s .

The v a l u e o f th e t o t a l c u r r e n t i s th e same i f we c a l c u l a t e fro m th e tű n n e li n g - H a m i l t o n i a n m eth o d 22 o r fro m th e p r e s e n t G re e n ’ s - f u n c t i o n m e th o d . However, we n e e d n o t a u t o m a t i c a l l y e x p e c t c o r r e s p o n d in g a g re e m e n t i n t h e c a s e o f th e e n e r g y d e n s i t y .23

8 . HIGHER-ORDER PROCESSES

A d ia g ra m te c h n iq u e i s p ro p o s e d i n S e c . 3 w h ic h i s s i m i l a r to th e one s u g g e s te d b y J o s e p h s o n .^ The t y p i c a l s t r u c t u r e o f th e n t h - o r d e r d ia g ra m s i s i l l u s t r a t e d i n F i g . 7 .

The c o n t r i b u t i o n o f th e f o u r t h - o r d e r d ia g ra m s to th e t u n n e l i n g c u r r e n t h a s b een c a l c u l a t e d b y S c h r i e f f e r a n d W ilk in s . 24 I n th e p r o c e s s e s c a l c u l a t e d by them one p a i r h a s b e e n b ro k e n u p and two e l e c t r o n s h av e t u n ­ n e le d th r o u g h th e b a r r i e r . S uch p r o c e s s e s w ere f i r s t o b s e r v e d by T a y lo r and B u r s t e i n ^ a s p e a k s i n t u n n e l i n g c h a r a c t e r i s t i c s . R e c e n tly , 25 2 A / n s t r u c t u r e h a s b e e n o b s e r v e d . 26 These p r o c e s s e s may b e i n t e r p r e t e d a s th e b r e a k i n g o f one p a i r and th e t u n n e l i n g o f n e l e c t r o n s i n th e same q u a n tu m -m e c h a n ic a l p r o c e s s .

We may a r g u e t h a t t h e r e i s no r e a s o n to s u p p o se t h a t t h e r e w ould be a g r e a t d i f f e r e n c e b e tw e e n th e a m p lit u d e s f o r th e b r e a k o f one o r s e v e r a l e l e c t r o n p a i r s i n p r o c e s s e s o f th e same o r d e r . The p r o p o s e d p r o c e s s i s th e b r e a k i n g o f p p a i r s and th e t u n n e l i n g o f n e l e c t r o n s . The v o l t a g e t h r e s h ­ o ld o f t h e s e p r o c e s s e s i s eV= 2 A C p / n ) > d u e to th e c o n s e r v a t i o n o f e n e r g y . The e x p e r i m e n t a l r e s u l t s o f R o c h l in and D o u g la s s ' may be i n t e r p r e t e d a s a

2 A ( p / n ~ ) s t r u c t u r e i n th e t u n n e l i n g c h a r a c t e r i s t i c s , a s i s d i s c u s s e d e l s e w h e r e .28

I t m u st b e s t r e s s e d t h a t th e p r o p o s e d m ethod i s n o t c o r r e c t to any o r d e r , a s th e f i r s t te rm o f th e G re e n ’ s f u n c t i o n i n th e i t e r a t i o n p r o c e ­ d u re g iv e n i n S e c , 3 c o n t a i n s u n p h y s i c a l c o r r e c t i o n s o f h i g h e r o r d e r t o th e lo w e s t n o n v a n is h in g o n e . I t seem s r e a s o n a b l e t o su p p o se t h a t i f we c o n s i d e r o n ly th e l e a d in g c o r r e c t i o n s o f th e h i g h e r - o r d e r p r o c e s s e s an d n e g l e c t th e u n p h y s i c a l c o r r e c t i o n s , we w i l l o b t a i n th e i n t e r e s t i n g c o n t r i b u t i o n s o f th e s e p r o c e s s e s . On th e o t h e r h a n d , th e a m p litu d e s o f th e h i g h e r - o r d e r c o r ­ r e c t i o n s s t r o n g l y d e c r e a s e a s th e o r d e r i n c r e a s e s . T h e r e f o r e th e p r o c e s s e s o f t h i s ty p e may be much more i n t e n s i v e i f th e - t r a n s i t i o n s o f th e e l e c t r o n s th r o u g h t h e b a r r i e r o c c u r a t some i m p e r f e c t i o n s o f th e b a r r i e r . I n th e l a s t c a s e o u r a p p ro a c h c a n n o t be a p p l i e d .

- 16

9 . CONCLUSION

A m any-body t r e a t m e n t o f th e t u n n e l i n g p r o c e s s e s h a s b e e n e l a b o r ­ a t e d . The p r e s e n t a p p r o a c h h a s d e a l t w i t h th e b e h a v i o r o f t h e e l e c t r o n s i n th e b a r r i e r , a s w e l l . The G re e n ’ s f u n c t i o n s have b e e n c a l c u l a t e d by an

i t e r a t i o n p r o c e d u r e . The c o n t r i b u t i o n s t o th e G re e n ’ s f u n c t i o n s i n th e lo w e s t a p p r o x im a tio n m ig h t be c l a s s i f i e d i n t o t h r e e d i f f e r e n t g r o u p s :

( 1 ) The t u n n e l i n g te rm <fGr w h ich d e s c r i b e s th e e l e c t r o n t r a n s i - t i o n th r o u g h th e b a r r i e r ' ;29

( 2 ) The r e n o r m a l i z a t i o n 't e r m cTGn и f o r th e m e ta l o n one o f th e s i d e s due to p r e s e n c e o f th e m e ta l on th e o p p o s i t e s i d e ; and

( 3 ) The te rm <fG$ due t o n o n o r t h o g o n a l i t y and n o n c o m p le te n e s s o f th e wave f u n c t i o n s . ^ T hese te rm s a r e d e p e n d e n t on th e c h o i c e o f th e s t e p f u n c t i o n s in t r o d u c e d i n S e c . 2 . These te r m s h av e b e e n n e g l e c t e d .

We h a v e fo u n d t h a t o n ly th e te rm o f th e f i r s t ty p e g i v e s c o n t r i b u ­ t i o n s t o th e c u r r e n t d e n s i t y i n th e b a r r i e r . We may c o n c lu d e t h a t o u r a p ­ p r o a c h c o m p u tin g th e c u r r e n t d e n s i t y i s c o r r e c t to o r d e r . The a d v a n ta g e o f w o rk in g w ith t h e c u r r e n t , d e n s i t y o c c u r s i n th e c o n s e q u e n t t r e a t m e n t o f th e e l e c t r o m a g n e t i c p r o p e r t i e s o f J o s e p h s o n j u n c t i o n s .

I n a few c a s e s , th e d e s c r i p t i o n o f th e e l e c t r o n s i n t h e r e g i o n o f th e b a r r i e r i s v e r y i m p o r t a n t . One o f them i s th e g e o m e t r i c a l r e s o n a n c e e f f e c t d i s c o v e r e d b y Tomasch 31 w here th e s u r f a c e and th e b o u n d a r y e f f e c t s p l a y im p o r ta n t r o l e s . T h is m ethod may b e v e r y p o w e r f u l i n th e d i s c u s s i o n o f . t h i s e f f e c t , b e c a u s e t h e b o u n d a ry a n d s u r f a c e e f f e c t s c o u ld b e ta k e n i n t o a c c o u n t i n th e s o l u t i o n o f th e s o - c a l l e d l e f t and r i g h t p ro b le m s and a d i r e c t c a l c u l a t i o n o f th e t u n n e l i n g c u r r e n t becom es p o s s i b l e u s i n g t h e s o l u t i o n s o f th e p a r t i c u l a r t h i n - f i l m p r o b le m s .

R e c e n tly some new t u n n e l i n g a n o m a lie s h a v e b e e n d i s c o v e r e d - ^ and

33 34

A n d erso n an d S u h l h a v e c a l l e d a t t e n t i o n to th e Kondo s c a t t e r i n g i n th e b a r r i e r a s t h e p o s s i b l e e x p l a n a t i o n o f t h i s e f f e c t . A p p e lb a u n r^ h a s c a l c u l a t e d th e t u n n e l i n g c u r r e n t u s in g th e t u n n e l i n g H a m i lto n i a n .R e c e n tl y Sólyom and th e a u t h o r h av e a p p l i e d t h i s G re e n ’ s - f u n c t i o n m eth o d t o t h i s p r o b l e m , ^ siumning up. a w id e c l a s s o f d ia g r a m s . The r e s o n a n t s c a t t e r i n g on th e p a r a m a g n e tic i m p u r i t i e s h a s b e e n t a k e n i n t o a c c o u n t by f i n d i n g th e s o l u ­ t i o n o f th e p a r t i c u l a r l e f t an d r i g h t p ro b le m s c o n s i d e r i n g a l s o th e p aram ag ­ n e t i c i m p u r i t i e s .

F i n a l l y , i t h a s b e e n c o n c lu d e d t h a t th e p h e n o m e n o lo g ic a l t u n n e l i n g H a m ilto n ia n c a n b e a p p l i e d t o th e c a l c u l a t i o n of t h e c u r r e n t i n th o s e c a s e s i n w h ich th e b a r r i e r e f f e c t s a r e n o t i m p o r t a n t .

ACKNOWLEDGMENTS

I s h o u ld l i k e t o th a n k P r o f e s s o r L. P á l f o r h i s c o n tin u o u s i n t e r e s t i n t h i s work an d P r o f e s s o r M o rre l H. Cohen f o r a s t i m u l a t i n g d i s c u s s i o n an d c o r r e s p o n d e n c e . I am v e r y g r a t e f u l t o K. L . N g a i, J .A . A ppelbaum , M. H.

C ohen, and J . C. P h i l l i p s f o r c o m m u n ic a tio n o f t h e i r work on th e f r e e e n e r g y o f t h e J o s e p h s o n j u n c t i o n p r i o r to i t s p u b l i c a t i o n a n d f o r p o i n t i n g o u t an e r r o r i n th e f i r s t v e r s i o n o f t h i s m a n u s c r i p t, t o P r o f e s s o r L .P . G o r’ kov f o r c a l l i n g th e p r o x i m i t y e f f e c t t o my a t t e n t i o n , a n d to D r. T.

W olfram f o r a s u g g e s tiv e d i s c u s s i o n on th e Tömasch e f f e c t . I w is h t o acknow­

le d g e num erous h e l p f u l d i s c u s s i o n s w i t h D r. C s. H a r g i t a i , D r. N. M enyhérd, a n d D r. J . S ólyom .

APPENDIX A

We h a v e d e r iv e d th e t u n n e l i n g te rm o f th e G re e n ’ s f u n c t i o n , b u t (2 6 ) c o n t a i n s th e s m o o th e d -o u t s t e p f u n c t i o n h( . We m u st show t h a t

<fGT

i s in d e p e n d e n t o f th e c h o ic e o f h( t o a good a p p r o x im a tio n . The f o ll o w i n g e x p r e s s i o n i s to be c a l c u l a t e d :

L < f / c f h c C y ) ] c f G T ( x , x ’) , c f y C B (A1)

o r t a k i n g th e F o u r i e r tr a n s f o r m w ith r e s p e c t t o th e tim e v a r i a b l e ,

Có’/ d h t (y)]<fGT' ( x , x ; E ) , if L/C В (

a

2 )

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

'< x . y;£)*((%- % У г т \ Ll ' C y , } (A3)

o r

f c / f x . y ; £ ) ( ( S „ -

Ъ ч) / г т ) с ;

( y .x '; £ ) - ( r — O }

O ) We e x p r e s s th e L a p l a c i a n o p e r a t o r s by th e i n v e r s e s o f th e G re e n ’ s f u n c t i o n s , u s i n g ( 8 ) and ( 9) :

A 4/ 2 m = - ( £ + ^ - V ( y ) ~ Г в ( у ; £ ) ) + G « 1 ( у , E ) (a5)

& y / 2m = - ( E + ^ - V ( y ) - £, -(</; £ ) ) + G~ f( y ; E) ,

an d

- 18 -

w here we made u se o f th e s t r u c t u r e o f th e m ass o p e r a t o r " ^ and th e a rro w above a d i f f e r e n t i a l o p e r a t o r i n d i c a t e s th e o p e r a n d .

I n s e r t i n g (A5) i n t o (A4) and u s i n g ((15)» we g e t th e f o l l o w i n g f o r m u l a :

[ó/(ff>c(y)]<fG r ' ( x y ; E ) = fGe'(x, y\ E)D' .(y ,x ‘)-De(x,y')Gr '(ii,x")} c}

= [cS/Jht ( y ^ ] c f G TiX} ( x , x / £ ) ,

( Аб)

w here we have i n t r o d u c e d th e new n o t a t i o n

<fGr>1> ( x , x ' t £ ) = f{G c'C x, l) ; E)Dr ( q , x ’) - D( ( y , x ) G ^ ( x , y ; E ) } x h l , ( y ) d 3y - ( r ^ ( ) .

■Л

We may c o n c lu d e t h a t 1 dGT c o u ld be c u t i n t o two p a r t s

• A / S XV t

<fGT —<fGT,D and d G Ti о . . The f i r s t te rm c o n t r i b u t e s d i r e c t l y to th e p h y s i c a l q u a n t i t i t e s e . g . , c u r r e n t d e n s i t y , b u t th e s e c o n d o n e i s

/\

s i m i l a r to cfGD , w h ich i s due t o th e b r e a k o f c o m p le te n e s s a n d o r t h o g o -

/ s

n a l i t y . I n S e c . 3 we n e g l e c t e d th e c o n t r i b u t i o n <fGB so we do t h e same

л Л

w i t h o G T D . F i n a l l y we c o n c l u d e . t h a t cfDT i s in d e p e n d e n t o f th e c h o ic e o f h ( i n t h i s a p p r o x im a tio n .

APPENDIX В

The t u n n e l i n g c u r r e n t can be o b t a i n e d by co m p u tin g e x p r e s s i o n s ( 4 1 a ) a n d (4 1 b ).

The G re e n ’ s f u n c t i o n s may be e x p r e s s e d by a c o m p le te s e t o f a p p ro ­ p r i a t e o n e - p a r t i c l e wave f u n c t i o n s

< * .* ') * E ХА л ^ С Гхо - ^ ) \ к ( х ) ( B1)

The s p e c t r a l r e p r e s e n t a t i o n o f th e n o rm al and an o m alo u s G re e n ’ s f u n c t i o n s a r e

G b , ; ( E ) : f d t e w ( i E c“ h

Фл*(Е)=[м exp(iEt)<fijliJ(t)= fdu> íL

^;----— +

—JOi*

— ^ вд (со )

у У t

t-CD+ie E - c o - i e

/

* * Л Е ) * Ф х * , . ( £ ) fB 3)

w here t - X0- Xo and

h có,* = { e x p f f t d c ^ - ^ J + l } ' 1 (B 4)

H ere th e s p e c t r a l f u n c t i o n s A x (со) and B x (oS) a r e r e a l . The c u r r e n t d e n s i t y may be e x p r e s s e d by t h e F o u r i e r tr a n s f o r m o f th e c a u s a l r e s p o n s e f u n c t i o n

Ji ( x ) = ! i m ( C - * R ) / d zf 4 , r f d t e x p ( i í t ) K c - ( x , x ' ) T l >

0 0

. (B

⁵

)

F—0 J $ J J-cxj

We c a l c u l a t e f i r s t th e o n e - p a r t i c l e c u r r e n t u s in g ( 3 7 ) and (Bi) - ( B 5) and i n t r o d u c i n g th e t r a n s i t i o n m a t r i x e le m e n ts

Ъ: = ^ \ ‘, e ^ X-X.r =~

an d

f \ , * ' ; r,C - j ' d 2f X'i> ^Xt \ ; i , r ,é ( * ^ =~ >, r , t ("вt)

s

The c u r r e n t d e n s i t y c a l c u l a t e d i n a s t r a i g h t f o r w a r d m an n er i s

fcuC*) = 2 /im (C -R ) e fd tex p Ш { Г .\ Тх> л . ЛЛ// х ) Gx'r (t)6y, e (~t) f a . ы ~ ( г ~ 0 }

~ 2 е П?о ^C'~R ') e { \ y 7 A’A; ift,r’ ^ 7x’*'» n 1(~ 2 7 Г Сх>г(Е'+ O Cx\ ( r ~ 0 J

= 2i l i m ( C - R ) e f £,

⁷

yjX ^Tx ,v, t ^{,n ( i d c u ]} f c / c J A

x 1

C c J )

F 0 I «/ ^

x —Z ^ u l ) - ( r ~ t ) J

\ E +

cú

- c j + ie E + c o - c o - i t y

^/

= 2 H i m ( C - R ) e f L Ty> s i h l r M T x y t fd c J A (< J ) id o l A y t ( c o ) ~ ~ ° " ' ‘r < r ~ t ) \ >

E~o ( A . X J J E t c o - a f + i e J

(B8) w here th e f a c t o r two i s due t o th e s p i n o r i e n t a t i o n s . F i n a l l y we have

j N, i M =~ /г,

^ту

, x : i , t , r ( * ) h .У,- r,tJ d < + > ^ A X ir(c J )J d c ^ A ^

1

( ° > ) - ~+ cj Z ^ j t ie j (B9)

- 20 -

o r

Í

ⁿ

,

Д * ) -

i.t.rWTx'^.r cjdoiAx r (bj)A)i-'l (cJ)(n0yiC-nCj,r')-

(B io ) The J o s e p h s o n c u r r e n t d e n s i t y can be c a l c u l a t e d i n a s i m i l a r way and th e r e s u l t i s

Jj,i ( x ) = - /f-e Im le x p {2 ie (c ,A<p-f-V)<() } r . T y y , Cr,(x)Tx y . rC fdco"Bx r (c j)jd c o B x - c (cd) x

1 Ä’x " ' " J ' J ’ ( B l l )

— - - ■

co--rJ =J$ ,v(x)sin[2e(c~'A<pi-Vx0) ] + J v(x)cos[2e(c'Ap +

И*0)/,

си -со t i e J 1,1

w here we have in t r o d u c e d t h e n o t a t i o n s ( 4 2 ) and ( 43) and th e a m p lit u d e s

Js v

an d Jc v .

At z e ro a p p l i e d v o l t a g e we h av e a much m ore s im p le e x p r e s s i o n o f th e J o s e p h s o n c u r r e n t

J

j

.

i

^esin[(2e/c)A<p]{ll^Tx_л .,■ ( r (x)Tx> V;r,jdcJB x>r(cd’)JdcdBx, e( с о ^B12^

w here /?to= пЫг= П(о[ and Jj Q > О

REFERENCES

J . B a rd e e n , P h y s . Rev. L e t t e r s 6 , 57 /1 9 6 1 / s 2» 1 ^7 / 1 9 6 2 / .

2 M.H. Cohen, L.M . F a li c o v , a n d J .C . P h i l l i p s , P h y s . R ev . L e t t e r s 8 , 516

/

1 9 6 2

/

" P .G . P ra n g e , P h y s . Rev. 1 5 1 . 1085 / 1 9 6 5 / 5 i n L e c t u r e s on th e Many-Body P ro b le m , e d i t e d b y E .R . C a i a n i e l l o /A cad e m ic P r e s s I n c . , New Y ork, 1964 , V o l. 2 .

^ P .G . de G en n es, P h y s. L e t t e r s £ , 22 / 1 9 6 5 / . 5 B .D . J o s e p h s o n , Adván. P h y s . 1 4 , 419 / 1 9 6 5 / .

8 W .J. Tomasch, P h y s . Rev. L e t t e r s 1£, 672 /1 9 6 5 /5 1 6 , 16 / 1 9 6 6 / j W .J. Tomasch a n d T. W olfram , i b i d . 16 , 552 / 1 9 6 6 / .

^ A .F .G . W y att, P h y s . Rev. L e t t e r s 1^., 401 /1 9 6 4 / .

8 J.M . R o w ell and Y .Y .L . S h e n , P h y s. R ev. L e t t e r s 1 8 , 2 15 / 1 9 6 6 / .

^ L .P . G orkov, Z h. E k sp e rim . i T e o r. F i z . 5 ib 755 / 1 9 5 8 / [ E n g lis h t r a n s l . t S o v i e t P hy s. - J E TP 11 , 696 / I 9 6 0 / ] ? A .A . A b rik o s o v , L .P . G o rk o v , and I . E . D z y a l o s h i n s k i i , M ethods o f Quantum! F i e l d T h e o ry i n S t a t i s t i c a l M ech an ics / P r e n t i c e - H a l l , I n c . , Englew ood C l i f f s , New’ J e r s e y , 1 9 6 5 /.

^ The s u f f i c i e n t c o n d i t i o n o f th e g iv e n fo rm o f E i s t h a t i t may be w r i t t e n i n t h e f o l l o w i n g form :

z = t s £ , * z B l w h ere and YLr a re th e p a r t s of th e m ass o p e r a t o r w h ich a r e w e ll l o c a l i z e d to t h e l e f t o r r i g h t s i d e s o f t h e b a r r i e r an d E B i s l o c a l i z e d to t h e b a r r i e r and i t s n e ig h b o rh o o d . F u r th e r m o r e , we h av e to s u p p o se t h a t

E e i s a l o c a l f u n c t i o n i n th e sp a c e

EB(x,x') = E s(x,x0-Xo)ó Y x-x').

T h is c o n d i t i o n i s f u l f i l l e d i f Ц d e s c r i b e s i n t e r a c t i o n o f e l e c t r o n s w ith i m p u r i t i e s c o r r e s p o n d in g to th e H a m ilto n ia n s

( M S ,

w here i s th e sP i n d e n s i t y and y C V'« i s th e e l e c t r o n d e n s i t y a t t h e p o s i t i o n o f th e i m p u r i t y w ith s p i n S . I n t h i s n o t a t i o n we may w r i t e

^ 'V '

E L = E C + Г.в and L r = E r + L B