Boundary Conditions for Magnetization in Thin Magnetic Elements

The magnetization dynamics of a magnetic element can be described using the Landau-Lifshitz equation of motion. This approach contains contributions from the non-uniform exchange interaction, as well as from the long-range dipole-dipole interaction, which is also nonuniform for non-ellipsoidal magnetic elements. The eigenfrequencies and eigenmode distributions of spin-wave excitations in such elements depend strongly on the boundary conditions at the element surfaces. Knowledge of these boundary conditions is important to calculate the spectra of magnetic linear excitations (spin waves) in the element, both in the case of a uniform ground state, when the element is magnetized by the external magnetic field to saturation, and in the case when the ground state is a strongly non-uniform (e.g. a magnetic vortex).