PHYSICAL REVIEW B VOLUME 49, NUMBER 6 1 FEBRUARY 1994-II Optimization of superconductor — normal-metal — superconductor Josephson junctions for high critical-current density A. Golub and B. Horovitz Department of Physics, Ben Gur-ion University of the Negev, P. O Box . 653, Beer Sh-eva 84105, Israel (Received 26 July 1993) The application of superconducting Bi,Sr2CaCu208 and YBa2Cu307 wires or tapes to electronic de- vices requires the optimization of the transport properties in Ohmic contacts between the superconduc- tor and the normal metal in the circuit. This paper presents results of tunneling theory in superconductor — normal-metal — superconductor (SNS) junctions, in both pure and dirty limits. We derive expressions for the critical-current density as a function of the normal-metal resistivity in the dir- ty limit or of the ratio of Fermi velocities and effective masses in the clean limit. In the latter case the critical current increases when the ratio y of the Fermi velocity in the superconductor to that of the weak link becomes much less than 1 and it also has a local maximum if y is close to 1. This local max- imum is more pronounced if the ratio of effective masses is large. For temperatures well below the criti- cal temperature of the superconductors the model with abrupt pair potential on the SN interfaces is con- sidered and its applicability near the critical temperature is examined. I. INTRODUCTION For many technical applications of superconductivity, a large critical current is an essential property. The in- tegration of superconducting devices in electronic cir- cuits requires that the current-carrying ability be preserved in electrode Ohmic contacts with metals, or at metallic-superconducting interfaces within the supercon- ducting device. Such contacts are present when the necessity for improved ductility of the superconducting ceramic material leads to the use of metal-matrix com- posites such as Bi2Sr2CaCu20s (BSCCO) granules dispersed in an Ag matrix, or BSCCO powder in an Ag tube. ' Composite wires or tapes can be modeled as a dispersion of strongly superconductive granular islands embedded in a metallic matrix such that the super- conducting order parameters are coupled by Joseph- son junctions. Each such junction is then a superconductor — normal-metal — superconductor (SNS) junction; the normal-metal part is also referred to as a weak link. The critical current of such SNS junctions determines the critical current of the composite system. It is therefore of considerable interest to identify which material parameters affect the SNS critical current and eventually optimize the Josephson junctions by the prop- erties (physical, chemical or geometrical) of the normal metal that could be adjusted to maximize the critical- current density. A direct Josephson effect in high-T, SNS junctions has been observed in many experiments (see review article for a list of references). The theoretical study of SNS- type Josephson junctions considers different temperature intervals below T, and various purity limits for materials which compose the SNS trilayer. For T close to T„SNS junctions were studied in the dirty limit, i.e. , in the case when the mean free paths of quasiparticles in the su- perconductor l, and in the weak link I„are less than the corresponding coherence lengths. A pure SNS sandwich with abrupt pair potential barriers at SN interfaces (valid for T « T, ) has been considered by Ishii at T=o and by Bardeen and Johnson. In this model the effective mass m„m„andFermi velocities U„v„ for the superconduc- tors and for the weak link, respectively, are the same. Near T, self-consistent solutions of the order parameter show that the pair potential is changed in a large region near the SN boundary, i.e. , the so-called proximity effect. This has a significant impact on the Josephson current affecting also its temperature dependence. The mode1 with abrupt pair potential barriers at SN interfaces needs then to be modified. The proximity effect of the pure SNS junction is known for v, =U„and m, =m„and to some extent for more general cases, ' The proximity effect with v, Wv„b tuwith m, =m„was considered by Kieselmann' who obtained numerically the pair poten- tial for several values of temperature. %e have used Gor'kov equations to calculate the critical Josephson current, though the approach based on the quasiclassical theory of superconductivity is possible. Quasiclassical equations and boundary conditions general enough to de- scribe a wide class of superconductors with magnetic ac- tive interfaces were obtained recently by Millis, Rainer, and Sauls. " This theory was generalized on superconductor — normal-metal double-layer system in the article. " In this work we evaluate the critical Josephson current of both pure and dirty SNS junctions in the present case where v, Wv„and m, Am„. The only assumption made is that the thickness d of the weak link is large compared with its coherence length, i.e. , weak tunneling. In Sec. II the pure SNS junction at temperatures near T, is studied by solving the integral equation for the order parameter and the critical current; the weak link is taken as a nor- mal metal with a critical temperature T, „=O. Optimum values of the parameters 0, /u„and m, /m„ for maximiz- 0163-1829/94/49(6)/4222(13)/$06. 00 49 4222 1994 The American Physical Society