Static properties of two- and three-dimensional superconducting constrictions

Abstract

Calculations have been performed on superconducting constrictions with hyperbolic geometry. Stationary Ginzburg-Landau equations are used, neglecting magnetic fields. Emphasis is placed on the difference between twoand three -dimensional constrictions, which is related to the difference between uniform-thickness (UT) and variable-thickness (VT) superconducting microbridges. The width of the constriction w, normalized to the coherence length ~, is indicated by the parameter A (---w/2sc). It is found that small (A < O. 1 ), three-dimensional constrictions and VT bridges have a sinusoidal current-phase relation, linear temperature dependence of the critical current Ic, and an IcR product (R is the normal state resistance) equal to the Ambegaokar-Baratoff expression for Josephson junctions near To. Twodimensional constrictions behave as if they consist of an inner core with junction properties, in series with the films on both sides. The core consists of the region within a coherence length from the center of the structure. This size is temperature dependent. The core shows a sinusoidal current-phase relation and LR according to Ambegaokar and Baratoff. For the whole constriction neither the phase difference nor R is finite. Two-dimensional constrictions have linear temperature dependence only when they are extremely narrow (A < 0. 001 ). In two-dimensional bridges the order parameter is depressed over a distance of approximately the coherence length ; in small three-dimensional constrictions this distance is approximately equal to the width. In narrow constrictions (and short microbridges) the current is not homogeneously distributed over the cross section. The effect has been investigated that occurs when in three-dimensional constrictions the width w is not much larger than lo, the electron mean free path in the basic material. To this purpose a Ginzburg-Landau equation is derived from the Zaitsev boundary conditions which is valid for continuously changing material parameters. The critical current is decreased, but the IeR product remains constant. The results of the calculations are compared with experimental results for superconducting microbridges.

Superconducting microbridges exhibiting Josephson properties

“…and three-dimensional bridges ( ~D B ) (Mooij and Dekker 1978). What is meant by these two geometrical configurations is illustrated in figure 9.…”