Spin-wave propagation in the presence of inhomogeneous Dzyaloshinskii-Moriya interactions

We theoretically investigate spin-wave propagation through a magnetic metamaterial with spatially modulated Dzyaloshinskii-Moriya interaction. We establish an effective Sch{\"o}dinger equation for spin-waves and derive boundary conditions for spin-waves passing through the boundary between two regions having different Dzyaloshinskii-Moriya interactions. Based on these boundary conditions, we find that the spin-wave can be amplified at the boundary and the spin-wave bandgap is tunable either by an external magnetic field or the strength of Dzyaloshinskii-Moriya interaction, which offers a spin-wave analogue of the field-effect transistor in traditional electronics.