  We calculate the Josephson current density j(phi) 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-reveresal 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.

Mesoscale and Nanoscale Physics