IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-19, NO. 3, MAY 1983 zyxwvuts J 629 zy DYNAMICS OF JOSEPHSON TUNNEL JUNCTIONS WITH A FINITE-WIDTH RIEDEL PEAK A. B. Zorin, K. K. Likharev and S. I. Turovets Department of Physics, Moscow S t a t e University, Moscow 117234, U.S.S.R. The basic properties of high-current-densityJoseph- son tunnel junctions are calculated from the microsco- pic theory taking into account a finite width 26 of t h e Riedelpeak. The calculated I-V curves for the finite values of thenormalizedcapacitance zyxwvutsrq 6 , the Riedel peak halfwidth zyxwvutsrq 6, and the parameter c1 zyxwvutsrqpo of pair current suppression are in good agreementwithpublishedexpe- rimental results for the high-current-density tunnel junctions Pb(1n)-oxide-Pb. The hysteresis parameter IR/Ic calculated from the microscopic theory as a fun- ction of thenormalizedcapacitance PC i s compared with the dependence following from the RSJ and the RSJN mo- dels. In addition, the low-frequency spectral density q(0) of the voltage fluctuations across, the junction is numericallyfoundand compared with that following fromthesimpleshot-noisemodel. 1. Introduction RecentadvancesintheJosephsontechnologyhavere- sulted in fabrication of small-area superconducting tu- nnel junctions with the critical current density up to 2X106 A/cm2 p2. These junctions have a very low RC ti- me c o n s t a n t , which makes them very suitable for a num- ber of applications. The theoretical analysis of dyna- mics of thesejunctionsis,however,complicateddueto t h e f a c t t h a t their McCumber-Stewart parameter 13, can be of the order or even less than unity. Thus thejun- ction intrinsic capacitance does not short out utterly their ac current, and the substantial ac component of the voltage across the junction should be taken into account. junctions has been studied using the relatively simple RSJ and RSJN models 3,4. Some features of real tunnel junctions (such as frequency dispersion of theJoseph- son current amplitude, shot and quantum fluctuations) arenot,however,takenintoconsiderationinthese mo- dels, The standard microscopic theory describes these features,butgives some sharp subharmonic-gap singula- rities on the I-V curves 6~7, in contrast with the real I-V curves where these singularities are rather smeared, In this paper we present the results of the junction dynamics analysis starting from the microscopic theory, for the case of finite width of the Riedel singulariti- e s i n all components of the tunnel current. Untilrecently,thedynamics of such,low-capacitance 2. Riedel Peak Broadening Acc0rdir.g to the microscopic theory of the Joseph- soneffect,thetunnelingcurrentthroughthesmall- area tunnel junction can be presented in the form where Qv=2e7/M is the Josephson oscillation frequency, V is theaveragevoltageacrossthejunction,and Wsz are the Fourier coefficients determined by thetimede- pendence of the phase difference @(t), The functions Iq(Q) and I zyxwvuts (a) are the complex ampli- P tudes of quasipartlcle and pair components, and can be Manuscript received November 30, 1982 expressed in terms of the unperturbed Green's functions of the superconducting electrodes. In the classical ve- rsion of the theory, all these amplitudes have singu- larities at Q=2A/$ owing to the gap singularity in BCS density of states. The singularities of the real parts are logarithmic (the singularity in ReIp(Q) is called a Riedel peak) and those of the imaginary parts are fi- nite discontinuous jumps (Fig.]). In reality, these singularities are somewhat broade- ned due to one or several of various possible mecha- nisms 8. The first class of the mechanisms (such as inelastic relaxation processes both in superconducting electrodes and in tunneling) leads to the Lorentzian shapeofthelogarithmicpeaks with a Kramers-Kronig-relatedsmearingofthedisconti- nuities where 6 is the relative halfwidth of the Riedel peak. The second class (for example, large-scale inhomogenei- ties of electrode materials) leads to the Gaussian shape of the broadened singularities sgn( x -1) -+ 2@{( -1)/u3-1, MQ zyxwvu m (4b) where @(x) is the error function and u2 is the dispersi- on of the Gaussian distribution. Figure 1 shows thecur- rent singularities broadened according to Eqs. (3),(4). We have found no e s s e n t i a l d i f f e r e n c e i n t h e results of our calculations using these two typesofthepeakbroa- dening,providedthewidths 6 and u are not large. The I-V curvescalculatedusingtheLorenzianshape(3) have turned out to be somewhat closer to the experimen- tal curves,andinthispaper we present the results for this case only. ." / / / / J 1 -3 -2 -I 0 I 2 3 (--I)/ +n S 2A Fig. 1. Broadening of the logarithmic peak (a) and the finite jump (b)ofthecurrentamplitudeswithLorenzi- an (dashed lines) and Gaussian (solid lines) shapes. 0018-9464/83/0500-0629$01.60 zyxwvu 0 1983 IEEE