Keldysh study of point-contact tunneling between superconductors

We revisit the problem of point-contact tunnel junctions involving one-dimensional superconductors and present a simple scheme for computing the full current-voltage characteristics within the framework of the nonequilibrium Keldysh Green function formalism. We address the effects of different pairing symmetries combined with magnetic fields and finite temperatures at arbitrary bias voltages. We discuss extensively the importance of these results for present-day experiments. In particular, we propose ways of measuring the effects found when the two sides of the junction have dissimilar superconducting gaps and when the symmetry of the superconducting states is not the one of spin-singlet paring. This last point is of relevance for the study of the superconducting state of certain organic materials like the Bechgaard salts and, to some extent, for ruthenium compounds.

Figure 1

Depiction of a set of frequencies involved in a multiple coherent tunneling process (the vertical axis corresponds to frequencies). The horizontal lines correspond to frequency-conserving tunneling processes that take place with amplitude $\mid t\mid$, while the vertical lines correspond to electron (hole) pair creation or destruction processes that occur with amplitude $\mid \Delta \mid$ (the superconducting gap is taken in this figure to have the same magnitude in the two sides of the junction). The chemical potentials, thermal distributions, and superconducting quasiparticle densities of states in the leads are schematically indicated. The lowest order nonzero contribution to the tunneling is depicted. The dashed lines and arrows indicate that the chain of interrelated higher order processes continues ad infinitum.

Figure 2

Zero temperature I-V characteristics of normal-superconductor junctions for both spin-singlet and spin-triplet paring. (a) N-S junctions for $t=0.2$ (lower curves) with and without applied magnetic field (dashed and solid lines, respectively, $h=0.2$) and $t=0.5$ (upper curve, solid line only since the effect of field is smaller and not displayed); the curves are vertically displaced for clarity. (b) N-T junctions for $t=0.2$ (upper curve) with and without applied magnetic field (dashed and solid lines, respectively, $h=0.2$) and $t=0.5$ (lower curve, solid line only).

Figure 3

Zero temperature I-V characteristics of S-T junctions with and without magnetic field (dashed and solid lines, respectively, $h=0.2$). The lower solid and dashed curves are for $t=0.2$ (without and with magnetic field) and the upper solid curve is for $t=0.5$. The dotted lines are (i) the straight unitary slope line for the reference N-N characteristics and (ii) the S-S characteristics for similar-parameters junctions (upper $t=0.5$ and lower $t=0.2$). The inset shows finite temperature characteristics of S-S junctions with different values of the left and right gap amplitudes. The dotted line is the reference N-N characteristics and the dashed line is the curve for ${\Delta }_2={\Delta }_1$. The solid lines correspond to ${\Delta }_2\ne {\Delta }_1$ (${\Delta }_2=3{\Delta }_1$ for the line closest to the dashed one and ${\Delta }_2=37∕3{\Delta }_1$ for the other one).

Figure 4

Zero temperature I-V characteristics of T-T junctions. The lower curve corresponds to $t=0.5$ and the upper one to $t=0.2$. The dotted line is the reference N-N characteristics. The inset shows finite temperature characteristics of T-T junctions with different values of the left and right gap amplitudes. We used the same value choices as in the inset of Fig. 3. Namely, the dotted line is the N-N characteristics and the dashed line corresponds to ${\Delta }_2={\Delta }_1$ while the solid lines are for ${\Delta }_2\ne {\Delta }_1$ (${\Delta }_2=3{\Delta }_1$ for the line closest to the dashed one and ${\Delta }_2=37∕3{\Delta }_1$ for the other one).