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Interface boundary conditions for dynamic magnetization and spin wave dynamics in a ferromagnetic layer with the interface Dzyaloshinskii-Moriya interaction

In this work we derive the interface exchange boundary conditions for the classical linear dynamics of magnetization in ferromagnetic layers with the interface Dzyaloshinskii-Moriya interaction (IDMI). We show that IDMI leads to pinning of dynamic magnetization at the interface. An unusual peculiarity of the IDMI-based pinning is that its scales as the spin-wave wave number. We incorporate these boundary conditions into an existing numerical model for the dynamics of the Damon-Eshbach spin wave in ferromagnetic films. IDMI affects the dispersion and the frequency non-reciprocity of the traveling Damon-Eshbach spin wave. For a broad range of film thicknesses L and wave numbers the results of the numerical simulations of the spin wave dispersion are in a good agreement with a simple analytical expression which shows that the contribution of IDMI to the dispersion scales as 1/L, similarly to the effect of other types of interfacial anisotropy. Suggestions to experimentalists how to detect the presence of IDMI in a spin wave experiment are given.