Spin waves can probe the Dzyaloshinskii-Moriya interaction (DMI) which gives rise to topological spin textures, such as skyrmions. However, the DMI has not yet been reported in yttrium iron garnet (YIG) with arguably the lowest damping for spin waves. In this work, we experimentally evidence the interfacial DMI in a 7~nm-thick YIG film by measuring the nonreciprocal spin wave propagation in terms of frequency, amplitude and most importantly group velocities using all electrical spin-wave spectroscopy. The velocities of propagating spin waves show chirality among three vectors, i.e. the film normal direction, applied field and spin-wave wavevector. By measuring the asymmetric group velocities, we extract a DMI constant of 16~$mu$J/m$^{2}$ which we independently confirm by Brillouin light scattering. Thickness-dependent measurements reveal that the DMI originates from the oxide interface between the YIG and garnet substrate. The interfacial DMI discovered in the ultrathin YIG films is of key importance for functional chiral magnonics as ultra-low spin-wave damping can be achieved.
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