  Periodically patterned metamaterials are known for exhibiting wave properties
similar to the ones observed in electronic band structures in crystal lattices.
In particular, periodic ferromagnetic materials are characterized by the
presence of bands and bandgaps in their spin-wave spectrum at tunable GHz
frequencies. Recently, the fabrication of magnets hosting Dzyaloshinskii-Moriya
interactions has been pursued with high interest since properties such as the
stabilization of chiral spin textures and nonreciprocal spin-wave propagation
emerge from this antisymmetric exchange coupling. In this context, to further
engineer the magnon band structure, we propose the implementation of magnonic
crystals with periodic Dzyaloshinskii-Moriya interactions, which can be
obtained, for instance, via patterning of periodic arrays of heavy-metals wires
on top of an ultrathin magnetic film. We demonstrate through theoretical
calculations and micromagnetic simulations that such systems show an unusual
evolution of the standing spin waves around the gaps in areas of the film that
are in contact with the heavy-metal wires. We also predict the emergence of
indirect gaps and flat bands and, effects that depend on the strength of the
Dzyaloshinskii-Moriya interaction. This study opens new routes towards
engineered metamaterials for spin-wave-based devices.
