  We theoretically propose a generation of an antiferro skyrmion crystal
(AF-SkX) in a synthetic bilayer antiferromagnetic system under an external
magnetic field. By performing the simulated annealing for a spin model on a
centrosymmetric bilayer triangular lattice, we find that the interplay among
the layer-dependent Dzyaloshinskii-Moriya interaction, the easy-plane
single-ion anisotropy, the interlayer exchange interaction, and the in-plane
magnetic field leads to instability toward the AF-SkX. The obtained AF-SkX
consists of the skyrmion layer and anti-skyrmion layer with opposite skyrmion
numbers, which results in no skyrmion number as well as spin scalar chirality
in the whole system. We also show that a ferri-type skyrmion crystal, where one
of the layers has the skyrmion number of one and the other does not, appears
when the magnitude of the magnetic field is varied. Our results provide a
guideline to experimentally search for the AF-SkX in bulk and layered
materials.
