Unidirectional spin wave propagation due to a saturation magnetization gradient

We demonstrate using micromagnetic simulations and a theoretical model that gradient in the saturation magnetization ($M_s$) of perpendicularly magnetized ferromagnetic film induces non-reciprocal spin wave propagation and, consequently an asymmetric dispersion relation. The $M_s$ adds linear potential to equation motion consistent with presence force. consider transformation from inertial reference frame which is constant accelerated where resulting force corresponds gradient. As Doppler effect, frequency shift leads Additionally, we show under certain circumstances, unidirectional waves can be achieved essential for design magnonic circuits. Our results become more relevant light recent experimental works suitable thermal landscape used dynamically modulate magnetization.