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Quantum theory of the small Josephson junction
Volume 25A. number 2
PHYSICS LETTERS
p a r a m e t e r s : the i n t e r a t o m i c d i s t a n c e d and the kind of divalent i m p u r i t y ~z2 = 518 d 2.20 and Za2 =357 d2.48 (Xand d i n / ~ . •r The m e a n e r r o r is about 90 A. A c c o r d i n g to L~ity [4] the divalent i m p u r i t i e s have no effect on the peak p o s i ti o n s of Z 1, so one gets ~ Z l = 893 d 1"67 with a mean e r r o r of about 50 A. To a f i r s t a p p r o x i m a t i o n the Z1 c e n t e r s fit a Mollwo line quite c l o s e l y (fig. 1); so one might conclude that a v e r y s y m m e t r i c model, a d e q u a tely d e s c r i b e s the Z l c e n t e r , depending only on the i n t e r a t o m i c d i s t a n c e . R e c e n t l y Smakula [7] obtained a b e t t e r fit to the e x p e r i m e n t a l data of the F - c e n t e r position, by using a r e l a t i o n s h i p of the above f o r m ; but in this c a s e the constants depend on the alkali ions in the c r y s t a l . A t h e o r e t i c a l t r e a t m e n t of this app r o a c h has been given by Wood [8]. We found a s i m i l a r r e l a t i o n (Smakula r e l a t i o n ) using t h r e e p a r a m e t e r s : the i n t e r a t o m i c d i s t a n c e , the d i v a lent i m p u r i t y {here Sr) and the kind of alkali ions
31 July 1967
neighbouring the c e n t e r (here K) ~Z2K= 402 d 2"41. This r e l a t i o n fits the e x p e r i m e n t a l data v e r y well (dashed line in the fig.); the m e a n e r r o r is about 10 A. It should be s t r e s s e d that only the m i x t u r e KC1-KBr d e v i a t e s by much m o r e {fig. 1), in a c c o r d a n c e with the co m p l ex b eh av i o u r of mixed c r y s t a l s [7]. The au t h o r s wish to acknowledge the i n t e r e s t and e n c o u r a g e m e n t of P r o f . E. L i i s c h e r .
References 1. E.Mollwo, Nachr. Ges. Wissen. G~ttingen {1931) p. 97. 2. H. Ivey, Phys. Rev. 72 (1947) 341. 3. J. Mort. Phys. Letters 21 {1966) 124. 4. F. Ltlty, Z. Phys. 182 (1964) 111. 5. K. Kojima, J. Phys. Soc. Japan 19 (1964) 868. 6. F. Seitz, Phys. Rev. 83 (1951) 134. 7. A. Smakula, N. Maynard, A. Repucci, Phys. Rev. 130 {1963) 113. 8. R.F. Wood, J. Phys. Chem. Solids 26 (1964) 615.
* * * * ~
QUANTUM
THEORY
OF
THE
SMALL
JOSEPHSON
JUNCTION
*
A. C. S C O T T
Dept. of Electrical Engineering. The University of Wisconsin. Madison. Wisconsin, USA Received 20 June 1967
lossless, small area Josephson junction is analyzed from the point of view of quantum dynamics. Schreedinger's equation in the flux representation reduces to Mathieu's equation.
T h e p h y s i c a l s y s t e m under c o n s i d e r a t i o n is shown in fig. 1. It c o n s i s t s of two s u p e r c o n d u c t ing p l a t e s s e p a r a t e d by a thin oxide l a y e r . We a s s u m e that the t h i c k n e s s of the oxide l a y e r , w, is s m a l l enough so a J o s e p h s o n c u r r e n t of s u p e r conducting e l e c t r o n s can tunnel through [1-3]. F u r t h e r m o r e , we a s s u m e that the t e m p e r a t u r e i s low enough so the l o s s e s a s s o c i a t e d with n o r ma l e l e c t r o n s can be n e g l e c t e d and a l s o that the a r e a A is s m a l l enough so the s e l f - f i e l d e f f e c t s can be ignored. The s y s t e m of fig. 1 can then be r e p r e s e n t e d by the equivalent c i r c u i t in fig. 2 w h e r e C is the c a p a c i t a n c e of the oxide l a y e r and * This work has been supported by the National Science Found ation. 132
the n o n l i n e a r inductor g i v e s the J o s e p h s o n c u r r e n t as a function of flux ~ = f v d t . i = /o sin(2e@/~)
(1)
The H a m i l t o n i a n for the s y s t e m can be w r i t t e n in t e r m s of the c h a r g e , Q in the c a p a c i t o r and the
I[,27;°;",',T:;,, Superconductor~/~
Oxide layer
Fig. 1. A small area Josephson junction.
Volume 25A, number 2
PHYSICS LETTERS
31 July 1967
Cooper p a i r s can p a s s through the junction, we find q = 4(a/coo)2 and a = 8[(a/coo) (E/~icoo) -
(a/coo)2].
po LNonlineor
inductance of ,Toeephlon current
o(..layer ° Capacltancn
Fig. 2. Equivalent nonlinear tank circuit. flux in the J o s e p h s o n inductor as
H = Q 2 / 2 C + (~iIo/2e) [1 - cos ( 2 e ~ /t /) ]
(2)
F o r s m a l l ~b, H .~ Q 2 / 2 C + ½(2e/o/~)d) 2 so the s m a l l s i g n a l i n d u c t a n c e i s t / / 2 e / o and we can d e fine a s m a l l s i g n a l r e s o n a n t f r e q u e n c y coo = d2eIo/PiC. We can now w r i t e S c h r o e d i n g e r ' s equation in the flux r e p r e s e n t a t i o n as ~2 a2~k q _ ~ I o
- 2 C O
2e~b
2e [1 - cos(--~-)] ~ =jr/ -~-
(3)
where f ~b2 ~ k * d ~ is the probability that the flux lies between the values ~bI and ~b2. A s s u m i n g ~k(~b,t) = u(qb) e x p ( - j E t / t i ) and m a k ing the substitution z = e O / ~ + ½ ~ , eq. ( 3 ) b e c o m e s M a t h i e u ' s equation, Mo d 2 u / d z 2 + ( 2 C / e ~ [ E - - ~ e (1 + cos 2z)] u : O, the so l u t i o n s of which have been studied in g r e a t de t a il . P u t t i n ~ i t into the s t a n d a r d f o r m of M c L a c h l a n [4], d u / d z 2 + (a - 2q cos 2z) u = O, and defining a = I o / 2 e as the m a x i m u m r a t e at which
Even in the s m a l l signal a p p r o x i m a t i o n t h e r e a r e two eigenfunctions c o r r e s p o n d i n g to each eigenfunction of the h a r m o n i c o s c i l l a t o r . Thus f o r the nth l e v e l of a s i m p l e h a r m o n i c o s c i l l a t o r [5] un ~ exp (-½z2)Hn (z) w h e r e H n is the nth o r d e r H e r m i t e polynomial. F o r M a t h i e u ' s e q u a tions the c o r r e s p o n d i n g eigenfunctions a r e u n ~ cen ( z, q) and Sen+ 1 ( z, q) w h e r e cen ( z, q) = ce n (z+Tr, q) and s e n (z, q) = -se n (z+Tr,q). T h e s e a r e evidently r e l a t e d to s y m m e t r i c and a n t i s y m m e t r i c combinations of the s i m p l e h a r m o n i c o s c i l l a t o r wavefunctions c e n t e r e d about z = ½7r and Z = 3t7. If q is l a r g e c o m p a r e d with unity, the allowed v a l u e s of a a r e given by the a s y m p t o t i c a p p r o x i mation [4, p.240] a n ~ - 2 q + (4n+2)vCqor E n (n+½)~w o as expected. F o r a given v al u e of q this a p p r o x i m a t i o n b e c o m e s p o o r e r with i n c r e a s i n g a orn. Ref. 4 i n d i c a t e s that the a s y m p t o t i c e x p r e s s i o n is in e r r o r by 1.7% at n = 2 for q = 160. Taking coo = 1010 r a d i a n s p e r second and a t y p i cal J o sep h so n c u r r e n t d e n s i t y of 16 A / e r a 2 i n d i c a t e s a c o r r e s p o n d i n g a r e a , A , of about 3 m i c r o n s × 3 m i c r o n s . S~naller a r e a s will lead to g r e a t e r d e viations f r o m the a s y m p t o t i c e x p r e s s i o n . 1. B.D.Josephson, Advan. Phys. 14 (1965) 419. 2. P.W. Anderson, Lectures on the many-body problem, ed. E. Caianello, (Academic Press) (1964) p. 113. 3. E. E. H. Shin and B. B. Schwartz, Phys. Rev. 152 (1966) 207. 4. N.W. McLachlan, Mathieu functions, (Oxford,1947). 5. J. C. Slater, Quantum theory of atomic structures I, (1960) p. 422.
133
PHYSICS LETTERS
p a r a m e t e r s : the i n t e r a t o m i c d i s t a n c e d and the kind of divalent i m p u r i t y ~z2 = 518 d 2.20 and Za2 =357 d2.48 (Xand d i n / ~ . •r The m e a n e r r o r is about 90 A. A c c o r d i n g to L~ity [4] the divalent i m p u r i t i e s have no effect on the peak p o s i ti o n s of Z 1, so one gets ~ Z l = 893 d 1"67 with a mean e r r o r of about 50 A. To a f i r s t a p p r o x i m a t i o n the Z1 c e n t e r s fit a Mollwo line quite c l o s e l y (fig. 1); so one might conclude that a v e r y s y m m e t r i c model, a d e q u a tely d e s c r i b e s the Z l c e n t e r , depending only on the i n t e r a t o m i c d i s t a n c e . R e c e n t l y Smakula [7] obtained a b e t t e r fit to the e x p e r i m e n t a l data of the F - c e n t e r position, by using a r e l a t i o n s h i p of the above f o r m ; but in this c a s e the constants depend on the alkali ions in the c r y s t a l . A t h e o r e t i c a l t r e a t m e n t of this app r o a c h has been given by Wood [8]. We found a s i m i l a r r e l a t i o n (Smakula r e l a t i o n ) using t h r e e p a r a m e t e r s : the i n t e r a t o m i c d i s t a n c e , the d i v a lent i m p u r i t y {here Sr) and the kind of alkali ions
31 July 1967
neighbouring the c e n t e r (here K) ~Z2K= 402 d 2"41. This r e l a t i o n fits the e x p e r i m e n t a l data v e r y well (dashed line in the fig.); the m e a n e r r o r is about 10 A. It should be s t r e s s e d that only the m i x t u r e KC1-KBr d e v i a t e s by much m o r e {fig. 1), in a c c o r d a n c e with the co m p l ex b eh av i o u r of mixed c r y s t a l s [7]. The au t h o r s wish to acknowledge the i n t e r e s t and e n c o u r a g e m e n t of P r o f . E. L i i s c h e r .
References 1. E.Mollwo, Nachr. Ges. Wissen. G~ttingen {1931) p. 97. 2. H. Ivey, Phys. Rev. 72 (1947) 341. 3. J. Mort. Phys. Letters 21 {1966) 124. 4. F. Ltlty, Z. Phys. 182 (1964) 111. 5. K. Kojima, J. Phys. Soc. Japan 19 (1964) 868. 6. F. Seitz, Phys. Rev. 83 (1951) 134. 7. A. Smakula, N. Maynard, A. Repucci, Phys. Rev. 130 {1963) 113. 8. R.F. Wood, J. Phys. Chem. Solids 26 (1964) 615.
* * * * ~
QUANTUM
THEORY
OF
THE
SMALL
JOSEPHSON
JUNCTION
*
A. C. S C O T T
Dept. of Electrical Engineering. The University of Wisconsin. Madison. Wisconsin, USA Received 20 June 1967
lossless, small area Josephson junction is analyzed from the point of view of quantum dynamics. Schreedinger's equation in the flux representation reduces to Mathieu's equation.
T h e p h y s i c a l s y s t e m under c o n s i d e r a t i o n is shown in fig. 1. It c o n s i s t s of two s u p e r c o n d u c t ing p l a t e s s e p a r a t e d by a thin oxide l a y e r . We a s s u m e that the t h i c k n e s s of the oxide l a y e r , w, is s m a l l enough so a J o s e p h s o n c u r r e n t of s u p e r conducting e l e c t r o n s can tunnel through [1-3]. F u r t h e r m o r e , we a s s u m e that the t e m p e r a t u r e i s low enough so the l o s s e s a s s o c i a t e d with n o r ma l e l e c t r o n s can be n e g l e c t e d and a l s o that the a r e a A is s m a l l enough so the s e l f - f i e l d e f f e c t s can be ignored. The s y s t e m of fig. 1 can then be r e p r e s e n t e d by the equivalent c i r c u i t in fig. 2 w h e r e C is the c a p a c i t a n c e of the oxide l a y e r and * This work has been supported by the National Science Found ation. 132
the n o n l i n e a r inductor g i v e s the J o s e p h s o n c u r r e n t as a function of flux ~ = f v d t . i = /o sin(2e@/~)
(1)
The H a m i l t o n i a n for the s y s t e m can be w r i t t e n in t e r m s of the c h a r g e , Q in the c a p a c i t o r and the
I[,27;°;",',T:;,, Superconductor~/~
Oxide layer
Fig. 1. A small area Josephson junction.
Volume 25A, number 2
PHYSICS LETTERS
31 July 1967
Cooper p a i r s can p a s s through the junction, we find q = 4(a/coo)2 and a = 8[(a/coo) (E/~icoo) -
(a/coo)2].
po LNonlineor
inductance of ,Toeephlon current
o(..layer ° Capacltancn
Fig. 2. Equivalent nonlinear tank circuit. flux in the J o s e p h s o n inductor as
H = Q 2 / 2 C + (~iIo/2e) [1 - cos ( 2 e ~ /t /) ]
(2)
F o r s m a l l ~b, H .~ Q 2 / 2 C + ½(2e/o/~)d) 2 so the s m a l l s i g n a l i n d u c t a n c e i s t / / 2 e / o and we can d e fine a s m a l l s i g n a l r e s o n a n t f r e q u e n c y coo = d2eIo/PiC. We can now w r i t e S c h r o e d i n g e r ' s equation in the flux r e p r e s e n t a t i o n as ~2 a2~k q _ ~ I o
- 2 C O
2e~b
2e [1 - cos(--~-)] ~ =jr/ -~-
(3)
where f ~b2 ~ k * d ~ is the probability that the flux lies between the values ~bI and ~b2. A s s u m i n g ~k(~b,t) = u(qb) e x p ( - j E t / t i ) and m a k ing the substitution z = e O / ~ + ½ ~ , eq. ( 3 ) b e c o m e s M a t h i e u ' s equation, Mo d 2 u / d z 2 + ( 2 C / e ~ [ E - - ~ e (1 + cos 2z)] u : O, the so l u t i o n s of which have been studied in g r e a t de t a il . P u t t i n ~ i t into the s t a n d a r d f o r m of M c L a c h l a n [4], d u / d z 2 + (a - 2q cos 2z) u = O, and defining a = I o / 2 e as the m a x i m u m r a t e at which
Even in the s m a l l signal a p p r o x i m a t i o n t h e r e a r e two eigenfunctions c o r r e s p o n d i n g to each eigenfunction of the h a r m o n i c o s c i l l a t o r . Thus f o r the nth l e v e l of a s i m p l e h a r m o n i c o s c i l l a t o r [5] un ~ exp (-½z2)Hn (z) w h e r e H n is the nth o r d e r H e r m i t e polynomial. F o r M a t h i e u ' s e q u a tions the c o r r e s p o n d i n g eigenfunctions a r e u n ~ cen ( z, q) and Sen+ 1 ( z, q) w h e r e cen ( z, q) = ce n (z+Tr, q) and s e n (z, q) = -se n (z+Tr,q). T h e s e a r e evidently r e l a t e d to s y m m e t r i c and a n t i s y m m e t r i c combinations of the s i m p l e h a r m o n i c o s c i l l a t o r wavefunctions c e n t e r e d about z = ½7r and Z = 3t7. If q is l a r g e c o m p a r e d with unity, the allowed v a l u e s of a a r e given by the a s y m p t o t i c a p p r o x i mation [4, p.240] a n ~ - 2 q + (4n+2)vCqor E n (n+½)~w o as expected. F o r a given v al u e of q this a p p r o x i m a t i o n b e c o m e s p o o r e r with i n c r e a s i n g a orn. Ref. 4 i n d i c a t e s that the a s y m p t o t i c e x p r e s s i o n is in e r r o r by 1.7% at n = 2 for q = 160. Taking coo = 1010 r a d i a n s p e r second and a t y p i cal J o sep h so n c u r r e n t d e n s i t y of 16 A / e r a 2 i n d i c a t e s a c o r r e s p o n d i n g a r e a , A , of about 3 m i c r o n s × 3 m i c r o n s . S~naller a r e a s will lead to g r e a t e r d e viations f r o m the a s y m p t o t i c e x p r e s s i o n . 1. B.D.Josephson, Advan. Phys. 14 (1965) 419. 2. P.W. Anderson, Lectures on the many-body problem, ed. E. Caianello, (Academic Press) (1964) p. 113. 3. E. E. H. Shin and B. B. Schwartz, Phys. Rev. 152 (1966) 207. 4. N.W. McLachlan, Mathieu functions, (Oxford,1947). 5. J. C. Slater, Quantum theory of atomic structures I, (1960) p. 422.
133