Frequency non-reciprocity of surface spin wave in Permalloy thin films

Surface spin waves in thin Permalloy films are studied by means of propagative spin wave spectroscopy. We observe a systematic difference of up to several tens of MHz when comparing the frequencies of counter-propagating waves. This frequency non-reciprocity effect is modeled using an analytical dipole-exchange theory that considers the mutual influence of non-reciprocal spin wave modal profiles and differences in magnetic anisotropies at the two film surfaces. At moderate film thickness (20 nm and below), the frequency non-reciprocity scales linearly with the wave vector and quadratically with the thickness, whereas a more complex non-monotonic behavior is observed at larger thickness. Our work suggests that surface wave frequency non-reciprocity can be used as an accurate spectroscopic probe of magnetic asymmetries in thin ferromagnetic films.