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Effect of microwave phonons on superconductive tunneling
Volume33A, number 5
EFFECT
OF
PHYSICS LETTERS
MICROWAVE
PHONONS
16 November 1970
ON S U P E R C O N D U C T I V E
TUNNELING
R. W. COHEN
RCA Laboratories, Princeton, N.J. 08540, ~ 4 Received 2 October 1970
We have calculated the effect of externally applied longitudinal microwave phonons on the complex energy gap of superconductors. The resulting change in the tunneling current of superconducting junctions accounts for some discrepancies between experiment and previous theory.
M e a s u r e m e n t s of the effect of e x t e r n a l l y a p p l i e d longitudinal m i c r o w a v e phonons on the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of s u p e r c o n d u c t ing junctions have r e s u l t e d in s e v e r a l i n t e r e s t i n g d i s c r e p a n c i e s [1, 2] between t h e o r y and e x p e r i ment. Although the change in the tunneling c u r r e n t 5I due to phonons i s d e s c r i b e d q u a l i t a t i v e l y by the t h e o r y [3, 4], the o b s e r v e d magnitude of 5I and i t s voltage and t e m p e r a t u r e dependence exhibit significant d e v i a t i o n s f r o m the p r e d i c t e d b e h a v i o r . The t h e o r e t i c a l a p p r o a c h [3, 4] to t h i s p r o b l e m has been to a s s u m e that the applied c o m p r e s s i o n a l wave g i v e s r i s e to a modulation of the e l e c t r o n i c e n e r g i e s through the d e f o r m a tion p o t e n t i a l i n t e r a c t i o n . The d i r e c t effect of the phonons on the e n e r g y gap A and the r e n o r r e a l i z a t i o n p a r a m e t e r Z of the s u p e r c o n d u c t o r s m a k i n g up the junction was neglected. We p r e sent h e r e the r e s u l t s of a c a l c u l a t i o n of the m o d i f i c a t i o n of A and Z due to m i c r o w a v e phonons and show that such a m o d i f i c a t i o n can a c count for s o m e of the o b s e r v e d [1,2] d i s c r e p a n c i e s between t h e o r y and e x p e r i m e n t . In g e n e r a l , the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c I(ID for a junction, whose n o r m a l state conductance i s ~n, i s given by [5]
effect c o n s i d e r e d h e r e , the a p p l i e d m i c r o w a v e phonons modify the A(i)(E) and, through eqs. (1) and (2), the tunneling c u r r e n t . We have c o m p u t e d the change 5 a(t) in the e n e r g y gap due to phonons under the following a s s u m p t i o n s : (1) E a c h s u p e r conductor i s i s o t r o p i c , h a s c r y s t a l l i n e d i m e n s i o n s l a r g e r than the phonon wavelength, and i s g o v e r n e d by the E l i a s h b e r g equations [6] of s u p e r c o n d u c t i v i ty, (2) the e x p e r i m e n t a l l y a p p l i e d m i c r o w a v e f i e l d r e s u l t s in an i s o t r o p i c d i s t r i b u t i o n of longitudinal phonons of a n g u l a r f r e q u e n c y 60o, and (3) in the f r e q u e n c y r a n g e of i n t e r e s t ~ o < Re A, the phonon s p e c t r u m of the s u p e r c o n d u c t o r s i s D e b y e like, and the e l e c t r o n - p h o n o n i n t e r a c t i o n i s d e s c r i b e d by a d e f o r m a t i o n potential i n t e r a c t i o n [e.g. 7]. With the e x c e p t i o n s of the a s s u m p t i o n s of gap and phonon i s o t r o p y , t h e s e a s s u m p t i o n s a r e e x p e c t e d to be v a l i d for the Pb f i l m s d e s c r i b e d in r e f s . [1] and [2]. Under the above a s s u m p t i o n s , the E l i a s h b e r g equations [6] m a y be r e a d i l y solved for 5A to f i r s t o r d e r in the phonon d e n s i t y (or m i c r o w a v e power). Dropping the s u p e r c r i p t s and setting P/= 1, we find 5A(E)
i(~Wo/Zo(E)){[Q(E+ ¢Oo)+ Q(E-¢Oo) ]
=
(3)
+co
I(V) : (~n f
dE 'N(1) (E)N(2) (E - V)[f(E-V) - f(E)]
-~
- (Ao(E)/E) [R(E+ 600) + R ( E - 60o)]} , (:)
w h e r e f(E) i s the F e r m i function d e t e r m i n i n g the occupancy of a q u a s i p a r t i c l e state of e n e r g y E, and
N(i)(E) = R e { E / [ E 2 - (A(i)(E))2] !/2}
where
Q(E),R(E) (2)
i s the e l e c t r o n i c d e n s i t y of s t a t e s of m e t a l i = 1,2, n o r m a l i z e d to the n o r m a l s t a t e value. The quantity A(Z)(E) i s the c o m p l e x e n e r g y gap. In the
= Re { ( A o ( E ) , E ) / [ E 2 - A 2 ( E ) ] l / 2 } .
(4)
The quantity A0(E) = [A01(E) + i A02(E) ] iS the e n e r g y gap of the u n p e r t u r b e d s u p e r c o n d u c t o r , and Z o i s the c o m p l e x pole r e n o r m a l i z a t i o n of the (unperturbed) q u a s i p a r t i c l e s . The quantity ~, i s a d i m e n s i o n l e s s coupling constant, p r o p o r t i o n a l to the a p p l i e d m i c r o w a v e p o w e r , 271
Volume 33A, n u m b e r 5
PHYSICS
9~ = ( ~ V s / 4 V F ) ( C S / Wo)2 .
272
16 November 1970
(5)
H e r e , vs i s the sound v e l o c i t y , v F i s the F e r m i velocity, C is the deformation potential and s is the strain produced by the applied sound wave. An e x a m p l e of t h e c h a n g e 61(V) = ( I ( V ) - I o ( V ) ) i n t h e c u r r e n t v o l t a g e c h a r a c t e r i s t i c of a j u n c t i o n m a d e up of i d e n t i c a l s u p e r c o n d u c t o r s , c a l c u l a t e d f r o m e q s . ( 1 ) - (5), i s s h o w n in fig. 1. T h e v a l u e s of t h e p a r a m e t e r s * g i v e n in t h e f i g u r e a p p r o x i m a t e t h o s e of P b a t 2 . 3 2 ° K w i t h W o / 2 n = 9.68 GHz. T h e c a l c u l a t e d 5 A(E) i s e s s e n t i a l l y p u r e i m a g i n a r a y a n d g i v e s r i s e to a 51(V) w h o s e f o r m i s q u a l i t a t i v e l y s i m i l a r to, b u t a p p r e c i a b l y s h a r p e r , t h a n t h e e x p e r i m e n t a l [ 1 - 4 ] 51(V). T h e e f f e c t of g a p a n i s o t r o p y [8] i n P b , w h i c h we h a v e i g n o r e d , would considerably broaden the calculated curves. T h e m a g n i t u d e of t h e e f f e c t c a n b e d e s c r i b e d in t e r m s of t h e c h a n g e 5N in t h e d e n s i t y of s t a t e s at E = A01(A01) , the g a p e d g e . D e n o t i n g b y r t h e r a t i o of the c o m p u t e d 5N to t h a t of t h e p r e v i o u s t h e o r y [3, 4], we f i n d a p p r o x i m a t e l y r = ( ~ V s / 4 ~ f 2 l Z o I V F ) ( ~ o3 7 2 / A ~ I 2 I A021), w h e r e A01 a n d h 0 2 a r e e v a l u a t e d at E = A01 , a n d it i s a s s u m e d t h a t yWo3/2/]Zo[A}(12 << {•02[ < Wo W e c o m p u t e * f o r lab t h e v a l u e s r ~ 0.2 at T =2.0°Kand r ~ 2 at T = 1.4°K. T h e r a p i d i n c r e a s e in r w i t h d e c r e a s i n g t e m p e r a t u r e , due to t h e d e c r e a s e * in I A02 [, i n d i c a t e s t h a t t h e d i r e c t e f f e c t of p h o n o n s on t h e e n e r g y g a p a c t u a l ly b e c o m e s l a r g e r t h a n t h e p r e v i o u s l y c o n s i d e r e d e n e r g y m o d u l a t i o n e f f e c t [ 3 , 4 ] at l o w t e m p e r a tures. The predicted magnitude and temperature d e p e n d e n c e of 6N a r e in g o o d a g r e e m e n t w i t h t h e r e s u l t s of r e f . [2]. In a d d i t i o n , in a g r e e m e n t w i t h e x p e r i m e n t [1], t h e e f f e c t d e s c r i b e d h e r e d o e s not v a n i s h f o r t h e c a s e of i d e n t i c a l s u p e r c o n d u c t o r s , a s i s p r e d i c t e d by t h e p r e v i o u s t h e o r y [4]. * The values ofAo2(Aol ) for Pb at various T were provided by B. Taylor. The o t h e r p a r a m e t e r s were obtained from ref. [6].
LETTERS 1
I
I
8 6 ~¢'-'- 4
5
r
I
I
T = 2.32°K AoI= 1.30 meV ~5 x I0- 4 meV Zo=2 'h O)o/e = 0 . 0 4 meV T = ~0-:'
f
F
-2 -4 2.54
I 2.56
I I 2.58 2.60 V (me V)
I 2.62
I 2.6,4
I 2.66
Fig. 1. Calculated change 51(V) in the tunneling c u r r e n t of a junction made up of identical s u p e r c o n d u c t o r s . The change 51 is n o r m a l i z e d to the u n p e r t u r b e d c u r r e n t Io(2A01). The p a r a m e t e r s * , approximating those of Pb, are given in the figure, and the value y = 10 -2 c o r r e sponds to the e x p e r i m e n t a l l y obtained power level o~s ~ 0.3 of ref. [1]. T h e a u t h o r w o u l d l i k e to t h a n k D r . Y. G o l d s t e i n for suggesting this work and for several very helpful discussions.
References [1] Y. Goldstein, B. Abeles and R. W. Cohen, Phys. Rev. 151 (1966) 349. [2] M. Cohen, Y. Goldstein and B. Abeies, Phys. Rev., to be published. [3] B. Abeles and Y. Goldstein, Phys. Rev. L e t t e r s 14 (1965) 595. [4] E. Lax and F. L. Vernon J r . . Phys. Rev. L e t t e r s 14 (1965) 256. [5] I. Giaever, Phys. Rev. L e t t e r s 5 (1960) 147, 464. [6] See for example R. D. P a r k s , Ed., Superconductivity (Marcel Dekker, New York 1969), Vol. 1, Ch. 10. [7] J. M. Ziman, E l e c t r o n s and phonons. (Oxford Univ e r s i t y P r e s s , London, England 1962), Ch. V. [8] A. J. Bennet, Phys. Rev. 140 (1965) 1902.
EFFECT
OF
PHYSICS LETTERS
MICROWAVE
PHONONS
16 November 1970
ON S U P E R C O N D U C T I V E
TUNNELING
R. W. COHEN
RCA Laboratories, Princeton, N.J. 08540, ~ 4 Received 2 October 1970
We have calculated the effect of externally applied longitudinal microwave phonons on the complex energy gap of superconductors. The resulting change in the tunneling current of superconducting junctions accounts for some discrepancies between experiment and previous theory.
M e a s u r e m e n t s of the effect of e x t e r n a l l y a p p l i e d longitudinal m i c r o w a v e phonons on the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of s u p e r c o n d u c t ing junctions have r e s u l t e d in s e v e r a l i n t e r e s t i n g d i s c r e p a n c i e s [1, 2] between t h e o r y and e x p e r i ment. Although the change in the tunneling c u r r e n t 5I due to phonons i s d e s c r i b e d q u a l i t a t i v e l y by the t h e o r y [3, 4], the o b s e r v e d magnitude of 5I and i t s voltage and t e m p e r a t u r e dependence exhibit significant d e v i a t i o n s f r o m the p r e d i c t e d b e h a v i o r . The t h e o r e t i c a l a p p r o a c h [3, 4] to t h i s p r o b l e m has been to a s s u m e that the applied c o m p r e s s i o n a l wave g i v e s r i s e to a modulation of the e l e c t r o n i c e n e r g i e s through the d e f o r m a tion p o t e n t i a l i n t e r a c t i o n . The d i r e c t effect of the phonons on the e n e r g y gap A and the r e n o r r e a l i z a t i o n p a r a m e t e r Z of the s u p e r c o n d u c t o r s m a k i n g up the junction was neglected. We p r e sent h e r e the r e s u l t s of a c a l c u l a t i o n of the m o d i f i c a t i o n of A and Z due to m i c r o w a v e phonons and show that such a m o d i f i c a t i o n can a c count for s o m e of the o b s e r v e d [1,2] d i s c r e p a n c i e s between t h e o r y and e x p e r i m e n t . In g e n e r a l , the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c I(ID for a junction, whose n o r m a l state conductance i s ~n, i s given by [5]
effect c o n s i d e r e d h e r e , the a p p l i e d m i c r o w a v e phonons modify the A(i)(E) and, through eqs. (1) and (2), the tunneling c u r r e n t . We have c o m p u t e d the change 5 a(t) in the e n e r g y gap due to phonons under the following a s s u m p t i o n s : (1) E a c h s u p e r conductor i s i s o t r o p i c , h a s c r y s t a l l i n e d i m e n s i o n s l a r g e r than the phonon wavelength, and i s g o v e r n e d by the E l i a s h b e r g equations [6] of s u p e r c o n d u c t i v i ty, (2) the e x p e r i m e n t a l l y a p p l i e d m i c r o w a v e f i e l d r e s u l t s in an i s o t r o p i c d i s t r i b u t i o n of longitudinal phonons of a n g u l a r f r e q u e n c y 60o, and (3) in the f r e q u e n c y r a n g e of i n t e r e s t ~ o < Re A, the phonon s p e c t r u m of the s u p e r c o n d u c t o r s i s D e b y e like, and the e l e c t r o n - p h o n o n i n t e r a c t i o n i s d e s c r i b e d by a d e f o r m a t i o n potential i n t e r a c t i o n [e.g. 7]. With the e x c e p t i o n s of the a s s u m p t i o n s of gap and phonon i s o t r o p y , t h e s e a s s u m p t i o n s a r e e x p e c t e d to be v a l i d for the Pb f i l m s d e s c r i b e d in r e f s . [1] and [2]. Under the above a s s u m p t i o n s , the E l i a s h b e r g equations [6] m a y be r e a d i l y solved for 5A to f i r s t o r d e r in the phonon d e n s i t y (or m i c r o w a v e power). Dropping the s u p e r c r i p t s and setting P/= 1, we find 5A(E)
i(~Wo/Zo(E)){[Q(E+ ¢Oo)+ Q(E-¢Oo) ]
=
(3)
+co
I(V) : (~n f
dE 'N(1) (E)N(2) (E - V)[f(E-V) - f(E)]
-~
- (Ao(E)/E) [R(E+ 600) + R ( E - 60o)]} , (:)
w h e r e f(E) i s the F e r m i function d e t e r m i n i n g the occupancy of a q u a s i p a r t i c l e state of e n e r g y E, and
N(i)(E) = R e { E / [ E 2 - (A(i)(E))2] !/2}
where
Q(E),R(E) (2)
i s the e l e c t r o n i c d e n s i t y of s t a t e s of m e t a l i = 1,2, n o r m a l i z e d to the n o r m a l s t a t e value. The quantity A(Z)(E) i s the c o m p l e x e n e r g y gap. In the
= Re { ( A o ( E ) , E ) / [ E 2 - A 2 ( E ) ] l / 2 } .
(4)
The quantity A0(E) = [A01(E) + i A02(E) ] iS the e n e r g y gap of the u n p e r t u r b e d s u p e r c o n d u c t o r , and Z o i s the c o m p l e x pole r e n o r m a l i z a t i o n of the (unperturbed) q u a s i p a r t i c l e s . The quantity ~, i s a d i m e n s i o n l e s s coupling constant, p r o p o r t i o n a l to the a p p l i e d m i c r o w a v e p o w e r , 271
Volume 33A, n u m b e r 5
PHYSICS
9~ = ( ~ V s / 4 V F ) ( C S / Wo)2 .
272
16 November 1970
(5)
H e r e , vs i s the sound v e l o c i t y , v F i s the F e r m i velocity, C is the deformation potential and s is the strain produced by the applied sound wave. An e x a m p l e of t h e c h a n g e 61(V) = ( I ( V ) - I o ( V ) ) i n t h e c u r r e n t v o l t a g e c h a r a c t e r i s t i c of a j u n c t i o n m a d e up of i d e n t i c a l s u p e r c o n d u c t o r s , c a l c u l a t e d f r o m e q s . ( 1 ) - (5), i s s h o w n in fig. 1. T h e v a l u e s of t h e p a r a m e t e r s * g i v e n in t h e f i g u r e a p p r o x i m a t e t h o s e of P b a t 2 . 3 2 ° K w i t h W o / 2 n = 9.68 GHz. T h e c a l c u l a t e d 5 A(E) i s e s s e n t i a l l y p u r e i m a g i n a r a y a n d g i v e s r i s e to a 51(V) w h o s e f o r m i s q u a l i t a t i v e l y s i m i l a r to, b u t a p p r e c i a b l y s h a r p e r , t h a n t h e e x p e r i m e n t a l [ 1 - 4 ] 51(V). T h e e f f e c t of g a p a n i s o t r o p y [8] i n P b , w h i c h we h a v e i g n o r e d , would considerably broaden the calculated curves. T h e m a g n i t u d e of t h e e f f e c t c a n b e d e s c r i b e d in t e r m s of t h e c h a n g e 5N in t h e d e n s i t y of s t a t e s at E = A01(A01) , the g a p e d g e . D e n o t i n g b y r t h e r a t i o of the c o m p u t e d 5N to t h a t of t h e p r e v i o u s t h e o r y [3, 4], we f i n d a p p r o x i m a t e l y r = ( ~ V s / 4 ~ f 2 l Z o I V F ) ( ~ o3 7 2 / A ~ I 2 I A021), w h e r e A01 a n d h 0 2 a r e e v a l u a t e d at E = A01 , a n d it i s a s s u m e d t h a t yWo3/2/]Zo[A}(12 << {•02[ < Wo W e c o m p u t e * f o r lab t h e v a l u e s r ~ 0.2 at T =2.0°Kand r ~ 2 at T = 1.4°K. T h e r a p i d i n c r e a s e in r w i t h d e c r e a s i n g t e m p e r a t u r e , due to t h e d e c r e a s e * in I A02 [, i n d i c a t e s t h a t t h e d i r e c t e f f e c t of p h o n o n s on t h e e n e r g y g a p a c t u a l ly b e c o m e s l a r g e r t h a n t h e p r e v i o u s l y c o n s i d e r e d e n e r g y m o d u l a t i o n e f f e c t [ 3 , 4 ] at l o w t e m p e r a tures. The predicted magnitude and temperature d e p e n d e n c e of 6N a r e in g o o d a g r e e m e n t w i t h t h e r e s u l t s of r e f . [2]. In a d d i t i o n , in a g r e e m e n t w i t h e x p e r i m e n t [1], t h e e f f e c t d e s c r i b e d h e r e d o e s not v a n i s h f o r t h e c a s e of i d e n t i c a l s u p e r c o n d u c t o r s , a s i s p r e d i c t e d by t h e p r e v i o u s t h e o r y [4]. * The values ofAo2(Aol ) for Pb at various T were provided by B. Taylor. The o t h e r p a r a m e t e r s were obtained from ref. [6].
LETTERS 1
I
I
8 6 ~¢'-'- 4
5
r
I
I
T = 2.32°K AoI= 1.30 meV ~5 x I0- 4 meV Zo=2 'h O)o/e = 0 . 0 4 meV T = ~0-:'
f
F
-2 -4 2.54
I 2.56
I I 2.58 2.60 V (me V)
I 2.62
I 2.6,4
I 2.66
Fig. 1. Calculated change 51(V) in the tunneling c u r r e n t of a junction made up of identical s u p e r c o n d u c t o r s . The change 51 is n o r m a l i z e d to the u n p e r t u r b e d c u r r e n t Io(2A01). The p a r a m e t e r s * , approximating those of Pb, are given in the figure, and the value y = 10 -2 c o r r e sponds to the e x p e r i m e n t a l l y obtained power level o~s ~ 0.3 of ref. [1]. T h e a u t h o r w o u l d l i k e to t h a n k D r . Y. G o l d s t e i n for suggesting this work and for several very helpful discussions.
References [1] Y. Goldstein, B. Abeles and R. W. Cohen, Phys. Rev. 151 (1966) 349. [2] M. Cohen, Y. Goldstein and B. Abeies, Phys. Rev., to be published. [3] B. Abeles and Y. Goldstein, Phys. Rev. L e t t e r s 14 (1965) 595. [4] E. Lax and F. L. Vernon J r . . Phys. Rev. L e t t e r s 14 (1965) 256. [5] I. Giaever, Phys. Rev. L e t t e r s 5 (1960) 147, 464. [6] See for example R. D. P a r k s , Ed., Superconductivity (Marcel Dekker, New York 1969), Vol. 1, Ch. 10. [7] J. M. Ziman, E l e c t r o n s and phonons. (Oxford Univ e r s i t y P r e s s , London, England 1962), Ch. V. [8] A. J. Bennet, Phys. Rev. 140 (1965) 1902.