Current-phase Relation of Sns and Ss1s Weak Links

Exact solutions of the current-phase relations and the critical currents of SS,S and SNS weak links are calculated for different boundary conditions and lengths of the S, and N regions. When the lengths of the S, and N re ions, 2d , are comparable to lcnl or smaller, both types of weak links behave similarly. When 2d >> fcnl, a SPS weak link behaves very much like a Josephson junction n but a SS,S weak link like a long superconducting wire with some of the Josephson-like properties remaining. WEAK LINKS.A weak link is a region which is in intimate contact with and located between two strongly superconducting regions and is able to transmit some of the long range superconducting properties between the strongly superconducting regions. The phase difference across the center region of a SS S weak link which is a superconductor In the superconducting state (T < T bulk of S,) is proportional to its length and current density. These solutions /1,2/ give rise to a sinusoidallike current-phase relationship for a short link which is fortuitously similar to that of a Josephson Junction. The SNS weak link center region is not in the superconducting state when not in contact with the outside superconductors. The center region aquires superconducting properties when sandwiched between the outside superconductors due to the proximity effect. The maximum loss free dc current is then the critical current of the SNS weak link. The essential difference between SS S and SNS weak links is that the phase difference sustained over the center region of a SNS weak link approaches the value ~ / 2 at the critical current density when 2dn/1 5,1 >> 1, whereas for a long SS,S weak link it is proportional to the length of center region 2d . n THEORY.An SNS weak link was treated analytically in /3/ and the results were compared with experiments by Clarke 141. Here we present exact numerical solutions for various material parameters and boundary conditions for both SNS and SSIS weak links. Definitions for the purpose of normalization are in /3/. Boundary conditions for a continuous current flow across the NS boundaries are /3/ XGs and X are the Gor'kov parameters /5/ which Gn are functions of temperature T and II and II n7 pectively. The parameter a determines whether the weak link is SS,S or SNS. If a > 0, then the center region of the weak link is a superconductor below the S, bulk transition temperature. For a < 0 the .renter region is denoted by N. SNS WEAK LINK.Since our SNS structure is assumed to be symmetric, it is only necessary to find solutions between x = 0 and x = x,. We assume that x = 10, unless otherwise specified, so that the Sregions approach the bulk limit (x, = 105 in conventional units). The value x is varied. The thick ness of the N-region, 2dn, is therefore varied. We find that there are at least two simple solutions of f and y for a fixed value of x and current density i which satisfy the boulidary conditions eqs. (1) and (2) and also df/dx = 0 at x = 0 and dy/dx = O a t x = x . 2 The total phase difference of the superconducting phase across the SNS or SS S sandwich is Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786251 where the second term on the right-hand side is the phase difference 2 ( ~ -XI) across the N or S1 region. The phase differences across the N-region only, which are very similar to the approximate results 131, approach those of the Josephson dc current phase relation in the very weak coupling limit (2d >> I) 161. This is not the case for a SS S n weak link. Figure 1 shows the total phase difference -2(x2-xO) across a SNS weak link as a function of current density i, calculated from eq. (3).