Auracher, P.

Richards, P.L.

Rochlin, G.I.

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Lawrence Berkeley National LaboratoryRecent WorkTitleOBSERVABILITY OF QUASIPARTICLE-PAIR INTERFERENCE CURRENT IN SUPERCONDUCTING WEAK LINKSPermalinkhttps://escholarship.org/uc/item/21f153q0AuthorsAuracher, P.Richards, P.L.Rochlin, G.I.Publication Date1972-11-01eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaSubmitted to Physical Review Letters ' ) ; PE·, ... E~v·-•\..:. ..... • t:.D IAWf<ENCf ~~' ·~, '., . ' 'll· .. .. -..~O:'J L.\30i~TORY . ~ '\ ' { i \ LBL-1197 Preprint ~. ' OBSERVABILITY OF QUASIPARTICLE-P_.AJR INTERFERENCE CURRENT IN SUPERCONDUCTING WEAK LIN1<S . F. Auracher, P. L. Richards, and G. I. Rochlin November 1972 Prepared for the U.S. Atomic Energy Commission under Contract W -7405-ENG-48 For Reference Not to be taken from this room DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. (I u 1 l.} .. .;:) Submitted to Physical Review Letters UNIVERSITY OF CALIFORNIA Lawrence Berkeley Laboratory Berkeley, California AEC Contract No. W-7405-eng-48 LBL-1197 OBSERVABILITY OF QUASIPARTICLE-PAIR INTERFERENCE CURRENT IN SUPERCONDUCTING WEAK LINKS F. Auracher,- P. L. Richards, and G. I. Rochlin November 1972 (I () ,j . i I,.J \) .. -1- LBL-1197 Observability of Quasiparticle-Pair Interference Current In Superconducting Weak Links . * F. Auracher Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, Calif. 94720 and P. L. Richardst and G. I. Rochlin Department of Physics, University of California and Inorganic Materials Research Division, Lawrence Berkeley Laboratory, Berkeley, Calif. 94720 ABSTRACT Calculations of static I -V curves, and of the heights 0 0 of steps induced by rf radiation, have been carried out for current biased Josephson junctions with both a phase-independ-ent quasiparticle conductance G and a phase-dependent quasi-. 0 particle pai~ "interference" conductance G1cos¢. Little or no effect of the term in G1co$¢ is found for junctions with zero capacitance. It thus appears that this term can be * t Research sponsored by the U.S. Army Research Office, Durham; Grant DA-ARO-D-31-124-70-G60 Research sponsored by the U.S. Office of Naval Research, Contract N00014-69-A-0200-1056. -2- LBL-1197 neglected in many practical applications of weak links. Sub-stantial effects are found for the case of finite junction capacitance. The term in c1cos¢ then enhances the hysteresis of the static I -V curve by an amount which roughly corre-o 0 sponds to a factor of 2-5 increase in capacitance, and causes the static voltage to increase above the value expected from the shunt conductance alone. The most easily observable ef-feet of the term in G1 cos¢ on the rf induced step heights is a pronounced shift in the values of rf current for which maxima occur for low values of normalized rf frequency. v· ,_ CJ -3- LBL-1197 1 According to the original work of Josephson as developed by 2 3 4 Josephson ' and by Nam, the current in a superconducting tunnel junction is given by I (t) Ic sin¢(t) + [G0(V) + G1(V) cos¢(t)]V(t). (1) The first term on the right is the usual phase dependent Josephson, or pair tunneling, current. The second term, G V is the dissipative . 0 quasiparticle tunneling current neglecting coherence.2 The third term,which is both dissipative and phase dependent, has been described5• 6 as an interference term between the pair and quasiparticle currents when the effects of coherence on the quasiparticle distribution are included.7 Because the quasiparticle current in a tunnel junction biased well below the gap is sniall, the interference term has usually been neglected. Recent experiments on the Josephson plasma resonance . 6 in Pb-Pb tunnel junctions by Pedersen et al. have shown evidence for the existence of the term in G1cos¢ and obtained a value for the ratio 5 A subsequent calculation by Poulson from the microscopic theory of tunnel junctions has shown that, for small voltages, the ratio is essentially independent of V, and is equal to -0.93 for the junctions used by Pedersen et al. It is of interest to explore whether a term in G1cos¢ with y of this order is present in the proper description of weak links other than tunnel junctions. Since the shunt conductance.G is large in point 0 contacts, Dayem bridges, and proximity effect bridges~ the existence of such a term might be of considerable practical importance for device LBL-1197 applications. The theory of such weak links is not developed sufficient-ly to answer this question reliably. In order to obtain· an experimental answer, we have ~lculated such experimentally observable quantities as the static I -V characteristic and the height of rf induced steps in 0 0 the presence of voltage-independent G1 and G0 with a constant negative ratio. In the absence of a shunt capacitance, little or no effect due to the y G V cos¢ term is obtained. When finite capacitance is intro-o duced, observable effects are predicted. None of the predicted effects, however, are dramatic enough for the existence of the term in y G cos¢ 0 to be verified from published data. We introduce the junction shunt capacitance into Eq. (1) in the usual way,8•9 letting I= C~~ for the capacitor, and replacing V with ~£t Using convenient dimensionless units8 we obtain, for the time 2e dt' evolution of the phase in a junction biased with a constant I , 0 d21h - dih S ~ i0 -sin¢- ~dT (l+y cos¢), ·c dT2 -(2) . 2 where i = I /I , T = 2e I t/hG , and S = 2ei C/hG . Taking G and 0 0 c. c 0 c c 0 0 y to be constant we have integrated Eq. (2) numerically to find the periodicity of d¢/dT and used this to determine the dependence of the IEf time averaged de voltage V 0 - G c ( dT )T across the junction on the applied 0 de current I for given values of y and S . 0 c Static I -V characteristics corresponding to earlier calculations 0 0 . 8 9 for y = 0 ' and also for y = -0.95 are shown in Fig. 1 for several values of the capacitance parameter S . These static characteristics c are independent of y for zero junction capacitance, as can be shown •• (J •,) J -5- LBL-1197 . 10 by analytic integration of Eq. (2) with S = 0. For a given c Be > 0, and a given bias current, the dc.voltage increases as y goes from 0 to -0.95. Note that for S and y both finite, the I -V C. 0 0 characteristic drops below the asymptotic line I = V G • This corre-o 0 0 2 sponds to a power dissipation greater than I /G over a wide range 0 0 -1/2 . of current. The voltage at which I = V G scales as S for 0 0 0 c 11 Be > 0.1. Similar calculations for y = -0.5 show smaller deviations, with the crossover moving towards higher currents; for the (probably unphysical) case of y > 0, the deviation has the opposite sense and the I -V curve bulges·· toward lower val tages. 0 0 It is difficult to establish the existence of the G cos¢ conduc-e tance by fitting data for various weak links to the curves in Fig. 1, since Be and G0 must be determined as well as y. The most pronounced effect of including the y G cos¢ conductance 0 is the increase in the hysteresis of the I -V curve at finite S . 0 0 c Defining a hysteresis parameter a= i . /i in the usual 'way, 8•9 we m1n o plot B vs a in Fig. 2 for y = 0, -0.5 and -0.95. For purposes of c comparison, we have also shifted the curve for y = 0 down by an amount corresponding to a reduction in B by a factor 0.45 to show that the c shape of the curve is most strongly affected in the range of small hysteresis (a~ l). The deviation of the static I -V curves for finite y can be most 0 0 easily understood by examining Fig. 3, in which we plot a single period of the time evolution of fixed value of i = 1.2. 0 the phase for various values of y and B c First consider the, casey = Be = 0. /For ¢ near n/2, the bias current flows primarily through the superconducting jqnction, so there is little at a -6- LBL-1197 dissipation in the shunt resistor. When ¢ is near 3TI/2, the junction current is opposite to the bias current, so the current (and dissipation) in the shunt resistor is large. As the phase spends more time near TI/2 during a cycle than near 3TI/2, the average dissipation is less than the 2 value I /G for the shunt alone. This corresponds to a time-averaged 0 0 forward supercurrent, and an average voltage V < I /G • . 0 0 0 As S is increased for y = 0, the phase-time characteristics change c as shown in Fig. 3(a). The inertial effect of the capacitance smooths out the ac supercurrent, equalizing the time spent near TI/2 and near 3TI/2. As S ~ oo at constant i , the voltage approaches the value V c 0 0 time-averaged supercurrent vanishes. I /G and the 0 0 lfuen we choose y = -0.95, we obtain the phase-time characteristics shown in Fig. 3(b). A careful comparison of Fig. 3(a) with Fig. 3(b) shows that the ratio of the time spent near TI/2 to that spent near 3TI/2 is in-dependent of y for S = 0. As we increase S from zero, the decreased con-e c ductance near ¢ 0 shortens the junction time constant and decreases the time spent near ¢ TI/2 while the increased time constant near ¢ = TI in-creases the time spent near ¢ = 3TI/2. There is a regime where the phase actually spends more time near 3TI/2 than near TI/2, giving a time-averaged reverse supercurrent, so that the dissipation and the time-averaged voltage are greater than for the conductance alone, V > I /G 0 0 0 As Sc ~ oo the capacitor again smooths out the time eyolution of the phase so that the I -V characteristic approaches the asymptotic value, V = I /G . 0 0 0 0 0 Because of the present interest in ac Josephson effect devices, the possible influence of a finite y on the ac response of various ( -7- LBL-1197 types of weak links is of practical importance. In order to explore this question we include an rf current source irf cosDrfT in the right-hand side of Eq. (2), where Drf = hwrfG0/2elc and i f = I f/I . As a measure of the rf response of the junction we have r r c computed the dependence of the height of the radiation induced steps with n = 0, 1 and 2 on irf" For B = 0, no effects of finite y were c observed for values of reduced frequency D < 0.16. The effects of rf ~ a finite y increase with D rf' but remain quite small even for D rf as large as 1.7. They include a small shift in the positions of the minima, a lifting of the minima away from zero step height, and a small decrease in the height of the_subsidiary maxima. All of these effects would be difficult to observe. Since junctions for applications general-ly have a = l, step heights were also computed for B = 0.15, the largest c value giving no hysteresis. The largest effects then occur for small values of the product D fB = w f C/G • Neither Be r c r o 0.15, nor y = -0.95 alone has an appreciable effect on the step heights, but in-combination they produce a marked shift in the positions of the subsidiary maxima, as shown in Fig. 4 for Drf = 0.16. Even though a finite y increases the harmonic content of the phase-time curve, the heights of integral order steps are not very sensitive to y. For these steps, the rf period averages over n periods of the phase oscillation. Although we have not explored the y~dependence of subharmonic steps, these might have a different sensitivity, as several rf cycles occur during one phase period. We conclude that a phase dependent quasiparticle-pair conductance term yG cos¢ produces potentially observable effects on the static 0 . . -8- LBL-1197 I - V curves and the radiation induced steps of Josephson junctions 0 0 shunted by both resistance and capacitance. In no case, however, are the effects large enough to be recognized in published data on weak links, or to require the inclusion of this term in an equivalent circuit used for device design. We gratefully acknowledge helpful discussions with D. N. Langenberg, D. J. Scalapino, andY. Taur. A portion of this work was performed under the auspices of the U. S. Atomic Energy Commission. ' ~ ... r J .. } J .:J j I ,j ,.) () ,,i .. ) ....) '··· -9- LBL-1197 REFERENCES 1. B. D. Josephson, Phys. Letters l• 251 (1962). 2. B. D. Josephson, Adv. in Phys. 14, 419 (1965). 3. B. D. Josephson, Rev. Mod. Phys. 1£, 216 (1964). 4. Sang Boo Nam, Phys. Rev. 156, 470 (1967). 5. U.K. Poulson, Phys. Letters 41A, 195 (1972). 6. N. F. Pedersen, T. F. Finnegan, and D. N. Langenberg, Phys. Rev. B (in press). 7. D. J. Scalapino, private communcation. 8. D. E. McCumber, J. Appl. Phys. ]i, 3113 (1968). 9. W. C. Stewart, Appl. Phys. Letters 11. 27l (1968). 10. We thank Y. Taur for first pointing out this result. 11. 1/2 Note that B =w C/G w T where w is the zero-phase plasma frequency. c p 0 p 0 p For w T > 1, the modulation of the phase in a cycle p 0 -depends on the ratio < d¢/dt } /w , becoming appreciable for p <d¢/dt > /w < 1. Since (V G /I )B 112 = w -l < d¢/dt >, the voltage p 0 0 0 c p at which crossing the asymptote V = I /G occurs will scale as 0 0 0 S -l/ 2 for all B ~ 1. c c FIGURE CAPTIONS Fig. 1. Normalized static I -V curves for various values of the 0 0 junction capacitance parameter S and pair quasiparticle interference c parameter, y. Fig. 2. Plot of the junction capacitance parameter S against the c junction hysteresis parameter a for various values of the pair-:-interference quasiparticle/conductance yG cos¢. The dashed line is they = 0 · . 0 curve shifted downward by an amount corresponding to a reduction in S by a factor of 0.45. c -10- LBL-1197 Fig. 3. Influence of a finite pair-quasiparticle interference conduc-tance yG cos~ on the time evolution of the junction phase ~. 0 (a) Zero interference current y 0, but various values of S . c (b) Fig. 4. Finite interference cuirent y = -0.95, and variouJ values of S . ~· c Plot of the height of the steps for n = 0, 1, and 2 as a func-tion of the rf current at the reduced frequency Orf = 0.16. The locations of the subsidiary maxima are shifted by the combined effects of a small junction capacitance S = 0.15 and a finite c quasiparticle-pair interference term y = -0.95. "0 Q) N 0 E ,_ 0 z ,, \.,i >) \ ~' -11- LBL-1197 --- y =-0.95 -----Y = o /~ /. 2 __ .,...., ------· t v t /. ./ ./ / Io c Normalized Voltage Fig. 1 /' ¢ YGo cos cp 2.0 Vo Go/ Ic XBL7211-7182 -12- LBL-1197 10 { y = 0 ---- {3~ = 0.45 f3c 0.2 0.4 0.6 0.8 1.0 « XBL 7211-7183 Fig. 2 ~ Q) 1/) 0 .t: Q_ c: 0 +-0 c: ::I J C} 3 2 17 0 3iT 2 7T -13-;Sc= 0 ---- f3c = 0.5 -·-·f3c = 2.0 .--R =0 /"" c . ----R =05 /""C . ---·f3c = 2.0 ~' ... ... ' " " " ~ , / I I I I (a) r = 0 • 'I /' • I /I • I /I • I / I • I / ~ . " / " . " / " . ... / ..... / .. ~ .... . "'~" .... /"" ........ /'/ ~ .. , , / , . I / I • I (b) r =-0.95 oL-~--~--~~---L--~--~~--~~ 0 0.5 1.0 Normalized Time (2eV0 /h) t XBL7211-7184 Fig. 3 LBL-1197 (.) 0.8 ........ ~ ........ +-~ .0\ 0.4 Q) I ~ Q) +-(f) ""0 Q) N 0 E ~ 0 z 0 LBL-1197 -14-!lrf = 0.16 f3c = 0.15, ----{f3c: 0.15, f3c- 0.0, y= ~0.95 y= .o.o y=- 0. 95 1.0 Normalized rf Current Fig. 4 2;0 Irtl Ic XBL 7211.;.7185 \ . J P-----------------LEGALNOTICE------------------~ This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights . TECHNICAL INFORMATION DIVISION LAWRENCE BERKELEY LABORATORY UNIVERSITY OF CALIFORNIA BERKELEY, CALIFORNIA 94720 ,..-

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OBSERVABILITY OF QUASIPARTICLE-PAIR INTERFERENCE CURRENT IN SUPERCONDUCTING WEAK LINKS

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OBSERVABILITY OF QUASIPARTICLE-PAIR INTERFERENCE CURRENT IN SUPERCONDUCTING WEAK LINKS

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