Dissipative boundary conditions for ferromagnetic resonance equations

Surface scattering of electrons and magnons results in nonuniform dissipation of the energy of ferromagnetic resonance (FMR) modes. This surface damping may be included in proper boundary conditions for the magnetic differential equations of motion, which are derived phenomenologically in the same way as the usual spin-pinning conditions. An analysis of the local energy balance in non-uniform modes leads to the concept of spin-wave energy flow; the dissipative boundary conditions describe the flow of the coherent mode energy into the surface. The magnitude of the surface damping parameters is only crudely estimated. Numerical calculations of the FMR linewidth and the shapes of standing-spinwave spectra indicate that experimentally significant effects may appear with damping parameters near to the estimated upper limits.