We study the interaction of surface acoustic waves with spin waves in ultra-thin CoFeB/Pt bilayers. Due to the interfacial Dzyaloshinskii-Moriya interaction (DMI), the spin wave dispersion is non-degenerate for oppositely propagating spin waves in CoFeB/Pt. In combination with the additional nonreciprocity of the magnetoacoustic coupling itself, highly nonreciprocal acoustic wave transmission through the magnetic film is observed. We systematically characterize the magnetoacoustic wave propagation in a thickness series of CoFeB($d$)/Pt samples as a function of magnetic field magnitude and direction, and at frequencies up to 7 GHz. We quantitatively model our results to extract the strength of the DMI and magnetoacoustic driving fields.
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