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Evidence for charge imbalance waves in phase-slip centers
Physica 107B (1981) 159-160 North-Holland Publishing Company
CE 1
EVIDENCE FOR CHARGE IMBALANCE WAVES IN PHASE-SLIP CENTERS
A.M. Kadin*, C. Varmazis**, and J.E. Lukens
Department of Physics State University of New York Stony Brook, NewYork 11794 We have examined the I-V characteristics o f superconducting phase-slip centers @SCs) located in the middle of narrow In channels. Features in dV/dl can be associated with resonances in charge imbalance waves driven by the PSC, as suggested by the model of Kadin, Smith and Skocpol. The velocity and temperature dependence of the waves thus inferred are in rough accord with theoretical predictions.
A PSC model which incorporates nonequilibrium dynamics was recently proposed by Kadin, Smith, and Skocpol (KSS) (I). In this transmission line model (Fig. la), the inductive channel represents the supercurrent and the resistive channel the normal current, the voltage between the two the nonequilibritrr potential (proportional to the charge imbalance), and the shunt conductance and capacitance allow for conversion between the two currents. The phase-slip process is approximated by an ideal Josephson element at the center. Since the parameters of the transmission line can be evaluated in terms of microscopic theory (1,2), the picture permits quantitative predictions regarding the nonequilibrium currents and voltages near a PSC. In addition, one can apply standard transmission line theory, involving the impedance and the propagation vector Zo= [(R+i~L)/(G+i~C)] , K = [ ~+i~L)(G+i~C)] ~
then to a good approximation the waves will be reflected at the banks with inversion, returning to the center with a phase lag AS= ~+4w£/~. This is equivalent in KSS to "shorting out" the ends of the transmission line at a distance £ from the center. The dc c u r r e n t r e s p o n s e I(V) o f a J o s e s p h s o n oscillator t o an e x t e r n a l s i g n a l a t ~J depends on t h e r e l a t i v e p h a s e s o f t h e e x t e r n a l and i n ternal oscillations. Therefore, varying V accross the PSC will cause ~j, k, and A8 to vary, and hence I will change correspondingly. In particular, small changes in the response of the PSC should be evident in dl/dV or dV/dI, enabling one to infer X by observing the features corresponding to A8 going through cycles of 2~. Q.)
p..(x)/e i ~(x)--~
For dc, KSS reproduces the results of the earli. er model of Skocpol, Beasley, and Tinkham (3), which have been adequately verified (4]. For ac, the Josephson oscillations in the center act as the driving force for waves at frequency ~j=2eV/h which propagate down the transmission line. These waves (see Pig. lb] are damped and dispersive, but in the high frequency limit have a velocity ~ v = I/(LC) = L2DAfT)/n] (D is the diffusion constant and A(T) the energy gap) and can be thought of as charge imbalance waves (CIWs], or alternatively can be identified with the propagating collective mode detected by Carlson and Goldman (5). Although there is some evidence for ac processes near PSCs (6], quantitative evidence for CIWs near PSCs is lacking. The idea behind our experiment was to provide a boundary to reflect the waves produced by a PSC back to the center, therebF using it as its own detector. One way to obtain this is with a narrow channel of length 2£ (much less than the ac decay length) which broadens suddenly to wide banks. A PSC oscillating in the middle of this channel sends out waves in both directions. If the width w<< k/4, where ~=v/~j is the wavelength. 03784363/81/0000~/$0250
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F i g . 1: a) R e p r e s e n t a t i o n o f t r a n s m i s s i o n l i n e model o f PSC. R, L, C, and G a r e r e s i s t a n c e , i n d u c t a n c e , c a p a c i t a n c e , and c o n d u c t a n c e p e r unit length, b) S p a t i a l d e p e n d e n c e o f normal and c o n d e n s a t e p o t e n t i a l s ~ and ~ a t an i n n s t a n t i n t i m e , showing damped wave~ e m a n a t i n g from t h e c e n t e r . 159
160
We tested these ideas on several samples consisting of In films about 1000 ~ thick, from about 0.5 to 4 p m long and about 0.3 u m wide, made using electron beam lithography and a lift-off technique. These samples did indeed exhibit a set of bumps in dV/dl in approximately the voltage range expected. An example is shown in Fig. 2 for a channel about 1.7 ~m long, for a number of temperatures near T =3.4 K. Prominent bumps are evident in the ~ange from 20 - 60 ~ V (vj from i0 - 30 GHz), and can be classified as three "peaks": Peak A is located near v~=12 GHz for all T. Peaks B and C, on the ot~er hand, move up in voltage (frequency~ as one moves farther from T , For both, ~T~e ~ near T and ~e ~ farther awa~ (E = I-T/Tc).o Over t~e whole range, 2~j(C) ~ 3~j(B). Features such as these could conceivably be due to something other than CIWs. One possibility is a microwave resonance, since the samples were coupled to a stripline in a microwave integrated circuit box. Such a resonance should be temperature independent, which might account for peak A, but not peaks B and C. On the other hand, suhharmoni~ gap structure (at V = 2A(T)/ne = 1800 pV/n e~ for In) are often sSen in short microhridges. However, because of the low voltages and incorrect e dependence of peaks B and C, their identification with subharmonics of the gap is highly unlikely. Furthermore, similar features tended to appear at higher voltages in shorter samples, which suggests a resonance in the channel. This leaves the possibility of CIW resonances. We carried out an approximate solution, using the lumped impedance equivalent of the finite length of transmission line (inset of Fig. 3) and parameters which correspond roughly to the experimental situation. Peaks in dV/dI occur when the condition A@ = 0 is satisfied, or 2£ = I/2, 3~/2, ... The voltages are t~en in the ratio 1:3:5. with a dependence %e ~ near T . As noted, peaks B and C have a similar depend- c ence, but go as 2:3. Obviously, the correspondence is not perfect. If however, we assume that peak B corresponds to the conditio~ 2Z=I/2, then Tor e=0.Ol we find v = Xvj= 5 x 10~cm/sec. For comparison, using an estimate of i000 ~ for the mean free path, ,the theoretical expression gives v = [2DA(T)/~r ~ = 9 × 10~cm/sec. This rough agreement, and consistent data on several other samples, suggests that even if the details of the ananlysis are probably incorrect, there is likely to he some truth in the underlying picture of CIW resonances. Both the model and the treatment of the boundary conditions are probably oversimplified, however, so that more work is needed to make this claim more convincing. An improved sample configuration might have a weakened spot in the center to "pin" the PSC, and thicker hanks, perhaps of a stronger superconductor, to help promote reflection. The channel might also he made of a superconductor with different parameters, such as AI.
To conclude, we have made preliminary measurements which tend to support the prediction that CIWs emanate from PSCs. *Now at School of Physics and Astronomy, U. of Minnesota, Minneapolis, Minn. 55455 **Now at Physics Dept., U. of Iraklion, Crete, Greece. Work supported by the Office of Naval Research.
REFERENCES ( i ) Kadin, A.M., Smith, L.N., and S k o c p o l , w . a . , J . Low Temp. Phys. 38 (1980) 497. (2) P e t h i c k , C . J . and S m i t h , H., Ann. Phys. (N.Y.) 119 (1979) 133; Artemenko, A.N., Volkov, A . F . , and Z a i t s e v , A.V., JETP L e t t . 27(1978) 113. (3) Skocpol, W.J., B e a s l e y , M.R., and Tinkham, M. J . Low Temp. Phys. 16 (1974) 145. (4) Dolan, G.J. and J a c k e l , L.D., Phys. Rev. L e t t . 34 (1977) 1628. (S) C a r l s o n , R.V. and Goldman, A.M., Phys. Rev. L e t t . 34 (1975) 11. (6) J i l l i e , D.W., Lukens, J . E . , and Kao, Y.H., IEEE T r a n s . Mag-13 (1977) 578; S k o c p o l , W.J. and J a c k e l , L.D., B u l l . Am. Phys. Soc. 26 (1981) 383.
dv dI
I Fig. 2: Set o f e x p e r i m e n t a l c u r v e s dV/dI vs. I f o r a r a n g e o f T n e a r T f o r an In c h a n n e l o f l e n g t h 1.7 ~m, w i d t h 0.~5 pm. S t a r t i n g w i t h t h e top c u r v e , T -T = 8, 19, 31, 42, 54 and 65 mK F r e q u e n c i e s uJ a~ peaks a r e i n d i c a t e d .
Fig. 3: Approximate solution of the model, using the lumped element equivalent for the transmission line of length 2£ (see inset), where Z = Z tanh(K£). O
CE 1
EVIDENCE FOR CHARGE IMBALANCE WAVES IN PHASE-SLIP CENTERS
A.M. Kadin*, C. Varmazis**, and J.E. Lukens
Department of Physics State University of New York Stony Brook, NewYork 11794 We have examined the I-V characteristics o f superconducting phase-slip centers @SCs) located in the middle of narrow In channels. Features in dV/dl can be associated with resonances in charge imbalance waves driven by the PSC, as suggested by the model of Kadin, Smith and Skocpol. The velocity and temperature dependence of the waves thus inferred are in rough accord with theoretical predictions.
A PSC model which incorporates nonequilibrium dynamics was recently proposed by Kadin, Smith, and Skocpol (KSS) (I). In this transmission line model (Fig. la), the inductive channel represents the supercurrent and the resistive channel the normal current, the voltage between the two the nonequilibritrr potential (proportional to the charge imbalance), and the shunt conductance and capacitance allow for conversion between the two currents. The phase-slip process is approximated by an ideal Josephson element at the center. Since the parameters of the transmission line can be evaluated in terms of microscopic theory (1,2), the picture permits quantitative predictions regarding the nonequilibrium currents and voltages near a PSC. In addition, one can apply standard transmission line theory, involving the impedance and the propagation vector Zo= [(R+i~L)/(G+i~C)] , K = [ ~+i~L)(G+i~C)] ~
then to a good approximation the waves will be reflected at the banks with inversion, returning to the center with a phase lag AS= ~+4w£/~. This is equivalent in KSS to "shorting out" the ends of the transmission line at a distance £ from the center. The dc c u r r e n t r e s p o n s e I(V) o f a J o s e s p h s o n oscillator t o an e x t e r n a l s i g n a l a t ~J depends on t h e r e l a t i v e p h a s e s o f t h e e x t e r n a l and i n ternal oscillations. Therefore, varying V accross the PSC will cause ~j, k, and A8 to vary, and hence I will change correspondingly. In particular, small changes in the response of the PSC should be evident in dl/dV or dV/dI, enabling one to infer X by observing the features corresponding to A8 going through cycles of 2~. Q.)
p..(x)/e i ~(x)--~
For dc, KSS reproduces the results of the earli. er model of Skocpol, Beasley, and Tinkham (3), which have been adequately verified (4]. For ac, the Josephson oscillations in the center act as the driving force for waves at frequency ~j=2eV/h which propagate down the transmission line. These waves (see Pig. lb] are damped and dispersive, but in the high frequency limit have a velocity ~ v = I/(LC) = L2DAfT)/n] (D is the diffusion constant and A(T) the energy gap) and can be thought of as charge imbalance waves (CIWs], or alternatively can be identified with the propagating collective mode detected by Carlson and Goldman (5). Although there is some evidence for ac processes near PSCs (6], quantitative evidence for CIWs near PSCs is lacking. The idea behind our experiment was to provide a boundary to reflect the waves produced by a PSC back to the center, therebF using it as its own detector. One way to obtain this is with a narrow channel of length 2£ (much less than the ac decay length) which broadens suddenly to wide banks. A PSC oscillating in the middle of this channel sends out waves in both directions. If the width w<< k/4, where ~=v/~j is the wavelength. 03784363/81/0000~/$0250
© No~h-HonandPublishingCompany
C
~slll/e
-4.o
-3.o
-;LO
!
-to
Illx)'-'"
o
×/AQ.
1o
2.0
3.0
4.0
F i g . 1: a) R e p r e s e n t a t i o n o f t r a n s m i s s i o n l i n e model o f PSC. R, L, C, and G a r e r e s i s t a n c e , i n d u c t a n c e , c a p a c i t a n c e , and c o n d u c t a n c e p e r unit length, b) S p a t i a l d e p e n d e n c e o f normal and c o n d e n s a t e p o t e n t i a l s ~ and ~ a t an i n n s t a n t i n t i m e , showing damped wave~ e m a n a t i n g from t h e c e n t e r . 159
160
We tested these ideas on several samples consisting of In films about 1000 ~ thick, from about 0.5 to 4 p m long and about 0.3 u m wide, made using electron beam lithography and a lift-off technique. These samples did indeed exhibit a set of bumps in dV/dl in approximately the voltage range expected. An example is shown in Fig. 2 for a channel about 1.7 ~m long, for a number of temperatures near T =3.4 K. Prominent bumps are evident in the ~ange from 20 - 60 ~ V (vj from i0 - 30 GHz), and can be classified as three "peaks": Peak A is located near v~=12 GHz for all T. Peaks B and C, on the ot~er hand, move up in voltage (frequency~ as one moves farther from T , For both, ~T~e ~ near T and ~e ~ farther awa~ (E = I-T/Tc).o Over t~e whole range, 2~j(C) ~ 3~j(B). Features such as these could conceivably be due to something other than CIWs. One possibility is a microwave resonance, since the samples were coupled to a stripline in a microwave integrated circuit box. Such a resonance should be temperature independent, which might account for peak A, but not peaks B and C. On the other hand, suhharmoni~ gap structure (at V = 2A(T)/ne = 1800 pV/n e~ for In) are often sSen in short microhridges. However, because of the low voltages and incorrect e dependence of peaks B and C, their identification with subharmonics of the gap is highly unlikely. Furthermore, similar features tended to appear at higher voltages in shorter samples, which suggests a resonance in the channel. This leaves the possibility of CIW resonances. We carried out an approximate solution, using the lumped impedance equivalent of the finite length of transmission line (inset of Fig. 3) and parameters which correspond roughly to the experimental situation. Peaks in dV/dI occur when the condition A@ = 0 is satisfied, or 2£ = I/2, 3~/2, ... The voltages are t~en in the ratio 1:3:5. with a dependence %e ~ near T . As noted, peaks B and C have a similar depend- c ence, but go as 2:3. Obviously, the correspondence is not perfect. If however, we assume that peak B corresponds to the conditio~ 2Z=I/2, then Tor e=0.Ol we find v = Xvj= 5 x 10~cm/sec. For comparison, using an estimate of i000 ~ for the mean free path, ,the theoretical expression gives v = [2DA(T)/~r ~ = 9 × 10~cm/sec. This rough agreement, and consistent data on several other samples, suggests that even if the details of the ananlysis are probably incorrect, there is likely to he some truth in the underlying picture of CIW resonances. Both the model and the treatment of the boundary conditions are probably oversimplified, however, so that more work is needed to make this claim more convincing. An improved sample configuration might have a weakened spot in the center to "pin" the PSC, and thicker hanks, perhaps of a stronger superconductor, to help promote reflection. The channel might also he made of a superconductor with different parameters, such as AI.
To conclude, we have made preliminary measurements which tend to support the prediction that CIWs emanate from PSCs. *Now at School of Physics and Astronomy, U. of Minnesota, Minneapolis, Minn. 55455 **Now at Physics Dept., U. of Iraklion, Crete, Greece. Work supported by the Office of Naval Research.
REFERENCES ( i ) Kadin, A.M., Smith, L.N., and S k o c p o l , w . a . , J . Low Temp. Phys. 38 (1980) 497. (2) P e t h i c k , C . J . and S m i t h , H., Ann. Phys. (N.Y.) 119 (1979) 133; Artemenko, A.N., Volkov, A . F . , and Z a i t s e v , A.V., JETP L e t t . 27(1978) 113. (3) Skocpol, W.J., B e a s l e y , M.R., and Tinkham, M. J . Low Temp. Phys. 16 (1974) 145. (4) Dolan, G.J. and J a c k e l , L.D., Phys. Rev. L e t t . 34 (1977) 1628. (S) C a r l s o n , R.V. and Goldman, A.M., Phys. Rev. L e t t . 34 (1975) 11. (6) J i l l i e , D.W., Lukens, J . E . , and Kao, Y.H., IEEE T r a n s . Mag-13 (1977) 578; S k o c p o l , W.J. and J a c k e l , L.D., B u l l . Am. Phys. Soc. 26 (1981) 383.
dv dI
I Fig. 2: Set o f e x p e r i m e n t a l c u r v e s dV/dI vs. I f o r a r a n g e o f T n e a r T f o r an In c h a n n e l o f l e n g t h 1.7 ~m, w i d t h 0.~5 pm. S t a r t i n g w i t h t h e top c u r v e , T -T = 8, 19, 31, 42, 54 and 65 mK F r e q u e n c i e s uJ a~ peaks a r e i n d i c a t e d .
Fig. 3: Approximate solution of the model, using the lumped element equivalent for the transmission line of length 2£ (see inset), where Z = Z tanh(K£). O