Nonreciprocal spin wave propagation in a ferromagnet with stepwise interfacial Dzyaloshinskii-Moriya interaction

In this paper, boundary conditions for Landau-Lifshitz equations at the interface between two ferromagnets with different interfacial Dzyaloshinskii-Moriya interaction are derived. We calculated and verified the analytical expression for the energy flux density continuity for the spin wave propagation through the ferromagnet with stepwise interfacial Dzyaloshinskii-Moriya interaction considering the boundary conditions mentioned. Analytical expressions for reflection, transmission coefficients and the nonreciprocity factor are calculated in the case of spin wave propagation through a ferromagnet the stepwise interfacial Dzyaloshinskii-Moriya interaction. Two different types of nonreciprocity effects for spin wave propagation through a ferromagnet with the stepwise interface Dzyaloshinskii-Moriya interaction are revealed for opposite orientations of Dzyaloshinskii-Moriya vector. The material parameters of a ferromagnet and the stepwise interfacial Dzyaloshinskii-Moriya interactions are found for which the extremely high nonreciprocity factor (> 10) is expected according to the results of calculations. The results of the paper deepen the knowledge about the spin wave propagation control in magnonic devices.