On the Selfconsistent Theory of Josephson Effect in Ballistic Superconducting Microconstrictions

TL;DR

This paper develops a microscopic, self-consistent theory for the Josephson effect in ballistic superconducting microchannels, analyzing how contact length influences the critical current near the critical temperature.

Contribution

It introduces a self-consistent integral equation approach for the order parameter in inhomogeneous microcontacts, extending the understanding of Josephson current behavior in ballistic regimes.

Findings

Critical current decreases with increasing contact length L.

For ultra-short channels, critical current corrections are sensitive to strong coupling effects.

Ballistic Sharvin resistance remains unchanged despite variations in critical current.

Abstract

The microscopic theory of current carrying states in the ballistic superconducting microchannel is presented. The effects of the contact length L on the Josephson current are investigated. For the temperatures T close to the critical temperature T_c the problem is treated selfconsistently, with taking into account the distribution of the order parameter $\Delta (r)$ inside the contact. The closed integral equation for $\Delta $ in strongly inhomogeneous microcontact geometry ($L\lesssim \xi_{0}, \xi_{0}$ is the coherence length at T=0) replaces the differential Ginzburg-Landau equation. The critical current $I_{c}(L)$ is expressed in terms of solution of this integral equation. The limiting cases of $L\ll \xi_{0}$ and $L\gg \xi_{0}$ are considered. With increasing length L the critical current decreases, although the ballistic Sharvin resistance of the contact remains the same as at L=0. For ultra short channels with $L\lesssim a_{D}$ ($a_{D}\sim v_{F}/\omega_{D}, \omega_{D}$ is the Debye frequency) the corrections to the value of critical current I_c(L=0) are sensitive to the strong coupling effects.