Interaction effects on proximity-induced superconductivity in semiconducting nanowires

We investigate the effect of electron-electron interactions on proximity-induced $s$-wave superconductivity in one-dimensional nanowires. We treat the interactions on a self-consistent mean-field level, and find an analytic expression for the effective pairing potential in the presence of interactions, valid for a weakly tunnel coupled wire. We show that for a set of two nanowires placed in parallel on a superconducting substrate, the interaction-induced reduction of the pairing energy could result in the effective interwire pairing potential exceeding the intrawire potential, which is one of the requirements for creating a time-reversal symmetric topological superconducting state in such a two-wire system.

Figure 1

Model system investigated in this work: Two semiconducting nanowires are placed in parallel on a superconducting substrate. The electrons in the wires are tunnel coupled to the superconductor, and the superconductor has an $s$-wave pairing, its order parameter being $\Delta$.

Figure 2

(a) Graphical presentation of numerical and analytic solutions to the self-consistent equation for $s$. The red connected dots are numerical evaluations of $(UT/\upsilon \Delta L_x){\sum }_{q,m}{\mathcal{G}}^{\mathrm{eh}}(q,i{\omega }_m)$. The green solid lines show our analytic approximation of the same expression. The blue dashed lines show $s/\upsilon$ for $\upsilon =1,0.1$. Self-consistent solutions for $s$ occur at intersections of the blue dashed lines with either the numerical or analytic curves. In both plots we used ${\mu }_{\mathrm{nw}}/\Delta =0.5$ and $T/\Delta ={10}^{-6}$. (b) Values for $\left|s\right|$ found numerically (red circles) and with analytic approximations (green crosses). (c) Resulting relative suppression of ${\Delta }_{1,2}$.