Destructive interference of direct and crossed Andreev pairing in a system of two nanowires coupled via an $s$-wave superconductor

We consider a system of two one-dimensional nanowires coupled via an $s$-wave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only direct pairing, where one wire is decoupled from the superconductor, or only crossed Andreev pairing, where each nanowire is considered as a spin-polarized edge of a quantum Hall state, we find that the gap is always reduced, by orders of magnitude in certain cases, when both types of pairing are present. Our analytical results are further supported by numerical calculations on a tight-binding lattice. Finally, we show that treating the proximity effect by integrating out the superconductor using the bulk Green's function does not reproduce the results of our exact diagonalization.

Figure 1

Geometry of considered model. A 2D conventional $s$-wave superconductor of width $d$ ($0<x<d$) separates two 1D nanowires. The system is infinite in the $y$ direction.

Figure 2

Excitation spectra for (a) $d=100{\xi }_s$, (b) $d={\xi }_s$, and (c) $d=0.01{\xi }_s$. For all plots, we choose ${\gamma }_R/{\gamma }_L=1.5$ and $k_F{\xi }_s=100$.

Figure 3

Proximity-induced gaps plotted as a function of superconductor width. (a) Analytical results for $k_F{\xi }_s=20$. Black curve corresponds to symmetric tunneling (${\gamma }_L={\gamma }_R$) [Eq. (11)], green curve corresponds to case of a single wire (${\gamma }_L=0$) [Eq. (12)], and blue curve corresponds to quantum Hall regime (${\gamma }_{L\uparrow }={\gamma }_{R\downarrow }\ne 0$ and ${\gamma }_{L\downarrow }={\gamma }_{R\uparrow }=0$) [Eq. (13)]. (b) Numerical results for $\mathrm{\Delta }=0.02t,{\mu }_s=0.3t,{\mu }_n=0.03t$, and $t_L=t_R=0.01t$, where $t$ is the hopping parameter in the superconductor. Inset: plot of $\mathrm{\Delta }{E}_g\equiv |E_g-(E_g^D-E_g^C)|$, showing very good quantitative agreement with Eq. (14).