Josephson effect in a Weyl SNS junction

We calculate the Josephson current density $j\left(\varphi \right)$ for a Weyl superconductor–normal-metal–superconductor junction for which the outer terminals are superconducting Weyl metals and the normal layer is a Weyl (semi)metal. We describe the Weyl (semi)metal using a simple model with two Weyl points. The model has broken time-reversal symmetry, but inversion symmetry is present. We calculate the Josephson current for both zero and finite temperature for the two pairing mechanisms inside the superconductors that have been proposed in the literature, zero-momentum BCS-like pairing and finite-momentum FFLO-like pairing, and assuming the short-junction limit. For both pairing types we find that the current is proportional to the normal-state junction conductivity, with a proportionality coefficient that shows quantitative differences between the two pairing mechanisms. The current for the BCS-like pairing is found to be independent of the chemical potential, whereas the current for the FFLO-like pairing is not.

Figure 1

(Left) Sketch of the geometry under consideration: An intrinsic Weyl (semi)metal for $-L/2<z<L/2$ is sandwiched between doped, superconducting Weyl metals for $z<-L/2$ and $z>L/2$. (Right) Schematic of the momentum space with two “Weyl cones” at momenta $\pm {\mathbf{K}}_0$. The angle between ${\mathbf{K}}_0$ and the $z$ axis is denoted $\alpha$.

Figure 2

Definition of the scattering matrices ${\mathcal{R}}_{+},{\mathcal{R}}_{-},{\mathcal{S}}_{++},{\mathcal{S}}_{+-},{\mathcal{S}}_{-+}$, and ${\mathcal{S}}_{--}$ for a Josephson junction.

Figure 3

Current-phase relationship for the Josephson current density $j\left(\varphi \right)$, normalized by the normal-state junction conductivity ${\sigma }_{\mathrm{N}}$. The top and bottom panels are for FFLO-like and BCS-like pairing, whereas the left and right panels are for the limits $\left|\mu \right|\gg \hslash v$ and $\left|\mu \right|\ll \hslash v$, respectively. The solid curves are for zero temperature; the dashed curves are for $k_{\mathrm{B}}T={\mathrm{\Delta }}_0$.

Figure 4

Critical current $j_{\mathrm{c}}$ normalized to normal-state junction conductivity ${\sigma }_{\mathrm{N}}$, as a function of $\mu L/\hslash v$. The top and bottom panels are for FFLO-like and BCS-like pairing, respectively. The different curves are for tempertures $k_{\mathrm{B}}T/{\mathrm{\Delta }}_0=0$, 0.2, 0.5, 0.7, and 1.0 (top to bottom inside each panel).