We present the results of a systematic micromagnetic study of the effect of the Dzyaloshinskii-Moriya interaction (DMI) on the spin wave band structure of two one-dimensional magnonic crystals (MCs), both with the same periodicity p = 300 nm, but different implementation of the DMI modulation. In the first system the artificial periodicity was achieved by modulating the interfacial DMI constant D, while in the second system also the sample morphology was modulated. Due to the folding property of the band structure in the dispersion relations of the magnonic crystals it is possible to extend the sensitivity of Brillouin light scattering towards weak DMI strength (D in the range from 0 to 0.5 mJ/m(2)), by measuring the frequency splitting of folded modes in high-order artificial Brillouin zones, since the splitting increases almost linearly with the band index. For relatively large values of the DMI (D in the range from 1.0 to 2.0 mJ/m(2)) the spin waves dispersion relations present flat modes for positive wavevectors, separated by forbidden frequency gaps whose amplitude depend on the value of D. These frequency gaps are more pronounced for the sample with morphology modulation. The non-reciprocal, localised, spatial profiles of these modes in both MCs are discussed with reference to spin waves in plain films and in isolated stripes of the same thickness.
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