Conductance of superconducting-normal hybrid structures

The dc conductance of normal-superconducting hybrid structures is discussed. It is shown that since the Bogoliubov-DeGennes (BDG) equation does not conserve charge, its application to create a Landauer-type approach for the conductance of the $NSN$ system is problematic. We “mend” this deficiency by calculating the conductance from the Kubo formula for a ring configuration, where charge conservation is imposed by the annular geometry. We show that the presence of a superconductor segment within an otherwise normal metal may reduce the overall conductance of the composite structure. This reduction enhances the tendency of the $NS$ composite to become insulating.

Figure 1

(Color online) Left panel: the “open” $NSN$ structure, where the perpendicular lines represent potential barriers (the explicit model treated in Sec. 4 assumes that the right interface is clean). Right panel: the corresponding ring $NSN$ system. The incoming and outgoing amplitudes of the electronlike and the holelike waves on both sides of the superconductor are marked by arrows.

Figure 2

(Color online) Left panel: conductance vs $d_S/\xi$ for several values of the transmission. Right panel: conductance vs the transmission for several values of $d_S/\xi$. Here the length of the $S$ region equals that of the $N$ region $\left(d_N=d_S\right)$. All information pertaining to values of $d_S$ smaller than $\xi$ is presented by dashed curves.

Figure 3

(Color online) Left panel: conductance vs $d_S/\xi$ for several values of the transmission. Right panel: conductance vs the transmission for several values of $d_S/\xi$. Here the length of the $S$ region is smaller than the length of the $N$ region $\left(d_N/d_S=9\right)$. All information pertaining to values of $d_S$ smaller than $\xi$ is presented by dashed curves.

Figure 4

(Color online) Comparison between the conductance Eq. (57) (dashed curves) and the conductance computed according to the Kubo formula, Eq. (17) (solid lines), as a function of $d_S/\xi$ for four values of the interface transmission. The latter conductance is always larger than the former.